Felix' Math Place (Posts about n-dimensional.)
https://math.fontein.de/tag/n-dimensional.atom
2019-11-17T10:38:25Z
Felix Fontein
Nikola
How to Obtain Reduction Maps for n-dimensional Infrastructures.
https://math.fontein.de/2009/07/21/how-to-obtain-reduction-maps-for-n-dimensional-infrastructures/
2009-07-21T05:43:54+02:00
2009-07-21T05:43:54+02:00
Felix Fontein
<div>
<div>
<p>
So far, we have seen how <a href="https://math.fontein.de/2009/07/20/n-dimensional-infrastructures/"><span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructures</a> can be defined. In the case of <a href="https://math.fontein.de/2009/07/20/one-dimensional-infrastructures/">one-dimensional infrastructures</a>, we saw that there is a (more or less) obvious way how to define a reduction map, which does not extend to the <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional case. We next want to motivate how a reduction map can be defined given <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="12" src="https://math.fontein.de/formulae/DxlmDzniFM_09XzQnKCoRu8h2JBxVd5JriHzmg.svgz" alt="d" title="d"></span>, using additional information which might be easier to obtain.
</p>
<p>
First, introduce on <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/bZVXuWhECmRV9pmTW_AgHZeLTOYax7svHPDCxQ.svgz" alt="\R^n" title="\R^n"></span> a lexicographic order as follows: for <span class="inline-formula"><img class="img-inline-formula img-formula" width="122" height="18" src="https://math.fontein.de/formulae/IyhZ8hrEm7aqvAEqrngKzppIjKZRSvA9dGYPRQ.svgz" alt="a = (a_1, \dots, a_n)" title="a = (a_1, \dots, a_n)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="117" height="18" src="https://math.fontein.de/formulae/lAyDYV3S69YtNSgsSQZTH_OUHNnZ5RTKYOf9sg.svgz" alt="b = (b_1, \dots, b_n)" title="b = (b_1, \dots, b_n)"></span>, define
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="409" height="18" src="https://math.fontein.de/formulae/AVOoUEwb3Di5uJI_Jl5bYJMqBgewR4QbUfmE7g.svgz" alt="a \le b :\Longleftrightarrow \exists i \in \{ 1, \dots, n \} : a_i \le b_i \wedge \forall j < i : a_i = b_i." title="a \le b :\Longleftrightarrow \exists i \in \{ 1, \dots, n \} : a_i \le b_i \wedge \forall j < i : a_i = b_i.">
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<p>
Note that this choice is rather random and can easily be replaced by other choices.
</p>
<p>
Assume that <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="15" src="https://math.fontein.de/formulae/EBTY1cww2.XbyZp0zeo5cPIgVso4OVcWdqrNBg.svgz" alt="\Lambda \subseteq \R^n" title="\Lambda \subseteq \R^n"></span> is a lattice, <span class="inline-formula"><img class="img-inline-formula img-formula" width="49" height="17" src="https://math.fontein.de/formulae/vroX4zy.EF6Jd8J4slYfcooFDuZqaHdjbpIajg.svgz" alt="X \neq \emptyset" title="X \neq \emptyset"></span> a finite set and <span class="inline-formula"><img class="img-inline-formula img-formula" width="111" height="18" src="https://math.fontein.de/formulae/g3EkzKjEux19bH4SPfjRnoZOQOaLVxwmoBd9LA.svgz" alt="d : X \to \R^n / \Lambda" title="d : X \to \R^n / \Lambda"></span> injective. Consider the projection <span class="inline-formula"><img class="img-inline-formula img-formula" width="119" height="18" src="https://math.fontein.de/formulae/XHL69cR1ss78J_iGymmCG4VcBZ941OC3CarIdg.svgz" alt="\pi : \R^n \to \R^n/\Lambda" title="\pi : \R^n \to \R^n/\Lambda"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="14" src="https://math.fontein.de/formulae/4f5VmTE4UWS69sDpvCVPYATBuG5Sf30s4CXuPw.svgz" alt="x \mapsto x + \Lambda" title="x \mapsto x + \Lambda"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="127" height="21" src="https://math.fontein.de/formulae/pu6d6sA2_z_m3zuQUAOTsV7FAmpUSsE6g4bcow.svgz" alt="\hat{X} := \pi^{-1}(d(X))" title="\hat{X} := \pi^{-1}(d(X))"></span>. Defining a function <span class="inline-formula"><img class="img-inline-formula img-formula" width="114" height="18" src="https://math.fontein.de/formulae/cxgoNn4N_cJ8PIFyQDOoobWed1pP9TpVDAUH1A.svgz" alt="\psi : \R^n / \Lambda \to X" title="\psi : \R^n / \Lambda \to X"></span> is the same as defining a function <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="20" src="https://math.fontein.de/formulae/mKNr924vboh5VrBlWp9TtJGPwO5acvma.v2_9g.svgz" alt="\varphi : \R^n \to \hat{X}" title="\varphi : \R^n \to \hat{X}"></span> which is invariant under <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/XY_2LK590TGO0.jb0ZbkmaiBi4kJr1xvp52q2g.svgz" alt="\Lambda" title="\Lambda"></span>, i.e. satisfies <span class="inline-formula"><img class="img-inline-formula img-formula" width="151" height="18" src="https://math.fontein.de/formulae/Sp.xXybT5sfo8dY_rtA4e2wZPQC.DLsZ67B.wA.svgz" alt="\varphi(t + \lambda) = \varphi(t) + \lambda" title="\varphi(t + \lambda) = \varphi(t) + \lambda"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="13" src="https://math.fontein.de/formulae/A5CppZBos4lcq9WsAE.53BBJujPwtmvA8McOnA.svgz" alt="\lambda \in \Lambda" title="\lambda \in \Lambda"></span>; in that case, we can set <span class="inline-formula"><img class="img-inline-formula img-formula" width="203" height="19" src="https://math.fontein.de/formulae/0wD3xtkjyB58XXYVH0VxtC6nZWxasY94HNCBUw.svgz" alt="\psi(t + \Lambda) := d^{-1}(\varphi(t) + \Lambda)" title="\psi(t + \Lambda) := d^{-1}(\varphi(t) + \Lambda)"></span>. Note that the condition <span class="inline-formula"><img class="img-inline-formula img-formula" width="90" height="16" src="https://math.fontein.de/formulae/UBqGzDw5g5LEA9_eZvt_oQ6OiaiyHcieA1CUUA.svgz" alt="\psi \circ d = \id_X" title="\psi \circ d = \id_X"></span> translates to <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="19" src="https://math.fontein.de/formulae/qkx6Zd7jqwa4iJCzP0j7vcrAA75YDNTngWaSxw.svgz" alt="\varphi|_{\hat{X}} = \id_{\hat{X}}" title="\varphi|_{\hat{X}} = \id_{\hat{X}}"></span>.
</p>
<p>
Hence, we have a discrete set <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="19" src="https://math.fontein.de/formulae/Q2uZMhKVGxQkovqEl2qMctJ1FvfM_0M6EeGTPA.svgz" alt="\hat{X} \subseteq \R^n" title="\hat{X} \subseteq \R^n"></span> which is invariant under translation by <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/XY_2LK590TGO0.jb0ZbkmaiBi4kJr1xvp52q2g.svgz" alt="\Lambda" title="\Lambda"></span>, and we want to define a function <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="20" src="https://math.fontein.de/formulae/mKNr924vboh5VrBlWp9TtJGPwO5acvma.v2_9g.svgz" alt="\varphi : \R^n \to \hat{X}" title="\varphi : \R^n \to \hat{X}"></span> satisfying <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="19" src="https://math.fontein.de/formulae/qkx6Zd7jqwa4iJCzP0j7vcrAA75YDNTngWaSxw.svgz" alt="\varphi|_{\hat{X}} = \id_{\hat{X}}" title="\varphi|_{\hat{X}} = \id_{\hat{X}}"></span>.
</p>
<p>
Both of the two sections which follow describe one way to obtain such <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="17" src="https://math.fontein.de/formulae/trC8ROiBHCvA1Mxjz0DMydgaaneA8jHc1Qwtww.svgz" alt="\hat{X}" title="\hat{X}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="11" src="https://math.fontein.de/formulae/G0SX86eTASn_9M49WF7HzDK85n7NoD6NrqM3ew.svgz" alt="\varphi" title="\varphi"></span>. The way describes in the second section fits perfectly for all totally real number fields <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>: think of <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/lZFl0.wtx.TkJAn.ie0DICOmVjwKyUYdr01UZA.svgz" alt="\Gamma" title="\Gamma"></span> as the image of the ring of integers <span class="inline-formula"><img class="img-inline-formula img-formula" width="28" height="15" src="https://math.fontein.de/formulae/Wj9qTigFv2JZS42zFIP4dypEZowSSDEtYLhlYw.svgz" alt="\calO_K" title="\calO_K"></span> under all embeddings <span class="inline-formula"><img class="img-inline-formula img-formula" width="167" height="16" src="https://math.fontein.de/formulae/roepyUgHZKtq6tZOvRYskaS5LnM7CyDiNl4FGg.svgz" alt="\sigma_1, \dots, \sigma_{n+1} : K \to \R" title="\sigma_1, \dots, \sigma_{n+1} : K \to \R"></span>, i.e.
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="288" height="18" src="https://math.fontein.de/formulae/g40YMEEa1bsUh7z.QvR_stL4CgmHKpV35znNFg.svgz" alt="\Gamma = \{ (\sigma_1(x), \dots, \sigma_{n+1}(x)) \mid x \in \calO_K \}." title="\Gamma = \{ (\sigma_1(x), \dots, \sigma_{n+1}(x)) \mid x \in \calO_K \}.">
</div>
<p>
The first section resembles more the general global field situation. The set <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> will consist of a finite set of ideals with bounded norms. The degree map will be the logarithm of the norm, and the <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="15" src="https://math.fontein.de/formulae/NSX3mGwNrOXj8Xc3q5z7NDr.jWDETMppkSC94A.svgz" alt="b_i" title="b_i"></span>'s correspond to the degrees of the infinite places.
</p>
</div>
<div class="subsection-block">
<h4>
Constructing a Reduction Map.
</h4>
<div>
<p>
In this section, we describe a way to construct a reduction map <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="11" src="https://math.fontein.de/formulae/G0SX86eTASn_9M49WF7HzDK85n7NoD6NrqM3ew.svgz" alt="\varphi" title="\varphi"></span>, given <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="17" src="https://math.fontein.de/formulae/trC8ROiBHCvA1Mxjz0DMydgaaneA8jHc1Qwtww.svgz" alt="\hat{X}" title="\hat{X}"></span>.
</p>
<p>
The main idea in the following is that if we want to define <span class="inline-formula"><img class="img-inline-formula img-formula" width="32" height="18" src="https://math.fontein.de/formulae/k1RD0iQX4eTrtI8VjjOpzF5Ybp4cta7_h0YSsQ.svgz" alt="\varphi(t)" title="\varphi(t)"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="158" height="18" src="https://math.fontein.de/formulae/1_2zJLGZZfbtkJRIuu8bW0SrTLuQr5D7dXzCDA.svgz" alt="t = (t_1, \dots, t_n) \in \R^n" title="t = (t_1, \dots, t_n) \in \R^n"></span>, to consider the area
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="298" height="18" src="https://math.fontein.de/formulae/zYwfVaa5aDpNOVyeosozJUYtHq0udziIN5WenA.svgz" alt="B_t := \{ (x_1, \dots, x_n) \in \R^n \mid \forall i : x_i \le t_i \}" title="B_t := \{ (x_1, \dots, x_n) \in \R^n \mid \forall i : x_i \le t_i \}">
</div>
<p>
and look at all elements <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="19" src="https://math.fontein.de/formulae/UpX192_wdjouteOA18YYu4YL7Am5gapdlu20IA.svgz" alt="\hat{X} \cap B_t" title="\hat{X} \cap B_t"></span>. By adding additional (numeric) information to every of these elements, one obtains an order (by comparing the additional information) which hopefully has a largest element, or a finite set of largest elements. From these largest elements, one chooses the largest one with respect to the lexicographic order <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="14" src="https://math.fontein.de/formulae/zlfN63Ix.fiq0WuW6KUdG.zhODns6kOCg.RygA.svgz" alt="\le" title="\le"></span> as <span class="inline-formula"><img class="img-inline-formula img-formula" width="32" height="18" src="https://math.fontein.de/formulae/k1RD0iQX4eTrtI8VjjOpzF5Ybp4cta7_h0YSsQ.svgz" alt="\varphi(t)" title="\varphi(t)"></span>.
</p>
<p>
To make this “additional information” more precise, we consider special functions <span class="inline-formula"><img class="img-inline-formula img-formula" width="98" height="20" src="https://math.fontein.de/formulae/SWb4G13xnVR9fKivn1YJzeKUQuXOkqvv_VQxCA.svgz" alt="\deg : \hat{X} \to \R" title="\deg : \hat{X} \to \R"></span> which should behave in a good way:
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<p>
A function <span class="inline-formula"><img class="img-inline-formula img-formula" width="98" height="20" src="https://math.fontein.de/formulae/SWb4G13xnVR9fKivn1YJzeKUQuXOkqvv_VQxCA.svgz" alt="\deg : \hat{X} \to \R" title="\deg : \hat{X} \to \R"></span> is said to be <em>reduction-inducing</em> if
</p>
<ol class="enum-level-1">
<li>
<p>
there exist real numbers <span class="inline-formula"><img class="img-inline-formula img-formula" width="105" height="16" src="https://math.fontein.de/formulae/CrmxGMFn81cmKA.w0DT0QkT3o412vGZtixRyWA.svgz" alt="b_1, \dots, b_n > 0" title="b_1, \dots, b_n > 0"></span> such that, for <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="17" src="https://math.fontein.de/formulae/enPpst21aU6YVDU5fiZSyG4cfkOWPVpNWiql7A.svgz" alt="x \in \hat{X}" title="x \in \hat{X}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="159" height="18" src="https://math.fontein.de/formulae/ZSWIPlhmwXNKD.tRsT0TNQxfnUpQR..om3PlJA.svgz" alt="\lambda = (\lambda_1, \dots, \lambda_n) \in \Lambda" title="\lambda = (\lambda_1, \dots, \lambda_n) \in \Lambda"></span>, we have
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="234" height="52" src="https://math.fontein.de/formulae/E6aPfAqa3jJcnryJySNccJigyvOcYsah_Y5HcQ.svgz" alt="\deg x + \sum_{i=1}^n b_i \lambda_i = \deg (x + \lambda);" title="\deg x + \sum_{i=1}^n b_i \lambda_i = \deg (x + \lambda);">
</div>
<p>
and
</p>
</li>
<li>
<p>
for every <span class="inline-formula"><img class="img-inline-formula img-formula" width="162" height="21" src="https://math.fontein.de/formulae/FwFOE_FVnp1vtYmRS4w0blHz6_noLVwqTNgYPA.svgz" alt="x = (x_1, \dots, x_n) \in \hat{X}" title="x = (x_1, \dots, x_n) \in \hat{X}"></span>, we have
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="465" height="21" src="https://math.fontein.de/formulae/WfXCiqnu4S.Gwzmsh8HLgNvLYMo7eO0bRv8ApA.svgz" alt="B_x := \{ x' = (x_1', \dots, x_n') \in \hat{X} \mid x_i' \le x_i, \; \deg x' > \deg x \} = \emptyset." title="B_x := \{ x' = (x_1', \dots, x_n') \in \hat{X} \mid x_i' \le x_i, \; \deg x' > \deg x \} = \emptyset.">
</div>
</li>
</ol>
</div>
</div>
<p>
Note that by this definition, there exist <span class="inline-formula"><img class="img-inline-formula img-formula" width="65" height="16" src="https://math.fontein.de/formulae/M5m8m5b27KDfE2Qt3lNEXgDCnPclpqAlUpF2SA.svgz" alt="a, A \in \R" title="a, A \in \R"></span> such that
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="267" height="52" src="https://math.fontein.de/formulae/6QhOh6X_g04dC.7VI6XfLSRvzvQr8hfwGNiANA.svgz" alt="a \le \deg (x_1, \dots, x_n) - \sum_{i=1}^n x_i b_i \le A" title="a \le \deg (x_1, \dots, x_n) - \sum_{i=1}^n x_i b_i \le A">
</div>
<p>
for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="162" height="21" src="https://math.fontein.de/formulae/FwFOE_FVnp1vtYmRS4w0blHz6_noLVwqTNgYPA.svgz" alt="x = (x_1, \dots, x_n) \in \hat{X}" title="x = (x_1, \dots, x_n) \in \hat{X}"></span>. Moreover, note that these functions with <span class="inline-formula"><img class="img-inline-formula img-formula" width="111" height="16" src="https://math.fontein.de/formulae/oKQ7iH1a58CmvAUcgm8oNHMFo1ZdAjlLlHvJPw.svgz" alt="a, A, b_1, \dots, b_n" title="a, A, b_1, \dots, b_n"></span> fixed correspond to functions <span class="inline-formula"><img class="img-inline-formula img-formula" width="131" height="19" src="https://math.fontein.de/formulae/em8.CnlyPf1BFGFAdgCE93KVKxXVa.is8lEDoQ.svgz" alt="\deg' : X \to [a, A]" title="\deg' : X \to [a, A]"></span> by
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="426" height="52" src="https://math.fontein.de/formulae/_jpFDOzH5MA3Jz.7wFO84763P_K84sF.Yt.VUQ.svgz" alt="\deg (x_1, \dots, x_n) = \deg' d^{-1}((x_1, \dots, x_n) + \Lambda) + \sum_{i=1}^n x_i b_i" title="\deg (x_1, \dots, x_n) = \deg' d^{-1}((x_1, \dots, x_n) + \Lambda) + \sum_{i=1}^n x_i b_i">
</div>
<p>
for <span class="inline-formula"><img class="img-inline-formula img-formula" width="162" height="21" src="https://math.fontein.de/formulae/FwFOE_FVnp1vtYmRS4w0blHz6_noLVwqTNgYPA.svgz" alt="x = (x_1, \dots, x_n) \in \hat{X}" title="x = (x_1, \dots, x_n) \in \hat{X}"></span>.
</p>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="98" height="20" src="https://math.fontein.de/formulae/SWb4G13xnVR9fKivn1YJzeKUQuXOkqvv_VQxCA.svgz" alt="\deg : \hat{X} \to \R" title="\deg : \hat{X} \to \R"></span> be a reduction-inducing function. For <span class="inline-formula"><img class="img-inline-formula img-formula" width="158" height="18" src="https://math.fontein.de/formulae/1_2zJLGZZfbtkJRIuu8bW0SrTLuQr5D7dXzCDA.svgz" alt="t = (t_1, \dots, t_n) \in \R^n" title="t = (t_1, \dots, t_n) \in \R^n"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="13" src="https://math.fontein.de/formulae/fBdpvOkdq9G_.BNOQcoePPSCj_EDabFXM2hiZQ.svgz" alt="\ell \in \R" title="\ell \in \R"></span>, consider
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="254" height="22" src="https://math.fontein.de/formulae/PzjrQ_LH5OundqCaAoGbJufBcMi9ruxOT15aSg.svgz" alt="B_{t,\ell} := \{ x \in \hat{X} \cap B_t \mid \deg x \ge \ell \}." title="B_{t,\ell} := \{ x \in \hat{X} \cap B_t \mid \deg x \ge \ell \}.">
</div>
<p>
Note that since <span class="inline-formula"><img class="img-inline-formula img-formula" width="173" height="20" src="https://math.fontein.de/formulae/4Z_IZJZeizfdTYcqBpq9GMuKzRwFsFjDbWdHUg.svgz" alt="\deg x \le A + \sum_{i=1}^n x_i b_i" title="\deg x \le A + \sum_{i=1}^n x_i b_i"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="202" height="21" src="https://math.fontein.de/formulae/GJ2is_w3cH8pj1bZ37FySoklYib7LNky1TA.yA.svgz" alt="x = (x_1, \dots, x_n) \in \hat{X} \cap B_t" title="x = (x_1, \dots, x_n) \in \hat{X} \cap B_t"></span>, and <span class="inline-formula"><img class="img-inline-formula img-formula" width="52" height="14" src="https://math.fontein.de/formulae/Cb6_nW4kgUzAGP_tB1WPFJRDMdAT8ClvJECjIg.svgz" alt="x_i \le t_i" title="x_i \le t_i"></span>, we see that <span class="inline-formula"><img class="img-inline-formula img-formula" width="30" height="17" src="https://math.fontein.de/formulae/Ir2H12G3eqXzB_PGMgc33eahKiFJ_zNzOjW5bg.svgz" alt="B_{t,\ell}" title="B_{t,\ell}"></span> is finite for every choice of <span class="inline-formula"><img class="img-inline-formula img-formula" width="7" height="12" src="https://math.fontein.de/formulae/TxqokZWmJwdp4cC9v4vThIX8YQePrhEiYeCmGw.svgz" alt="\ell" title="\ell"></span>. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="137" height="20" src="https://math.fontein.de/formulae/GW2iGxgKaWKaP9op4JqHuMebkNv2QyWEHo0vkw.svgz" alt="A + \sum_{i=1}^n t_i b_i < \ell" title="A + \sum_{i=1}^n t_i b_i < \ell"></span>, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="18" src="https://math.fontein.de/formulae/MrXHgesVi5ZoHxhuQDrXTL9rrZjH7jdX7_q2rQ.svgz" alt="B_{t,\ell} = \emptyset" title="B_{t,\ell} = \emptyset"></span>, and as <span class="inline-formula"><img class="img-inline-formula img-formula" width="49" height="17" src="https://math.fontein.de/formulae/vroX4zy.EF6Jd8J4slYfcooFDuZqaHdjbpIajg.svgz" alt="X \neq \emptyset" title="X \neq \emptyset"></span> we get <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="18" src="https://math.fontein.de/formulae/67FtAzGpaRzwJFMbEpm8cDPDPaNbzJZK6adT0A.svgz" alt="\abs{B_{t,\ell}} \to \infty" title="\abs{B_{t,\ell}} \to \infty"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="66" height="14" src="https://math.fontein.de/formulae/vdKK0.94K1sI9Rx1xBynj6LLEXatt6gPx5QIKQ.svgz" alt="\ell \to -\infty" title="\ell \to -\infty"></span>. Hence, <span class="inline-formula"><img class="img-inline-formula img-formula" width="201" height="18" src="https://math.fontein.de/formulae/wUT7xQ6YpRioNUWgSekxuXD45uL4IwlS2hA1EQ.svgz" alt="\ell(t) := \max\{ \ell' \mid B_{t,\ell'} \neq \emptyset \}" title="\ell(t) := \max\{ \ell' \mid B_{t,\ell'} \neq \emptyset \}"></span> exists. Then, define <span class="inline-formula"><img class="img-inline-formula img-formula" width="154" height="20" src="https://math.fontein.de/formulae/riUVQAqDTHrkdZRQZ88IQXLiAgvJtgzaZt8SPw.svgz" alt="\varphi(t) := \max_{\le} B_{t,\ell(t)}" title="\varphi(t) := \max_{\le} B_{t,\ell(t)}"></span>.
</p>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/Y7EdX0PEWGWK0fXtRMoCbkISGE2gGeQ1BLN.CQ.svgz" alt="C \in \R" title="C \in \R"></span> be a constant such that for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="158" height="18" src="https://math.fontein.de/formulae/1_2zJLGZZfbtkJRIuu8bW0SrTLuQr5D7dXzCDA.svgz" alt="t = (t_1, \dots, t_n) \in \R^n" title="t = (t_1, \dots, t_n) \in \R^n"></span>, we have
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="237" height="52" src="https://math.fontein.de/formulae/ZN76IctaNoGTLOSOLkPGqPwI6TVGxcVcbH9F3Q.svgz" alt="B_{t,\ell} \neq \emptyset \quad \text{for } \ell := \sum_{i=1}^n t_i b_i + C." title="B_{t,\ell} \neq \emptyset \quad \text{for } \ell := \sum_{i=1}^n t_i b_i + C.">
</div>
<p>
Note that since <span class="inline-formula"><img class="img-inline-formula img-formula" width="27" height="16" src="https://math.fontein.de/formulae/fj9Neom9DqzFph0HOWPyJl4KwpcIbtCRRNHl6A.svgz" alt="\deg" title="\deg"></span> is reduction-inducing, a maximal such <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/759yRxgUGr_PnVCdKXtyozLox.ROnoDZRDRPww.svgz" alt="C" title="C"></span> exists.
</p>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
For <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="17" src="https://math.fontein.de/formulae/enPpst21aU6YVDU5fiZSyG4cfkOWPVpNWiql7A.svgz" alt="x \in \hat{X}" title="x \in \hat{X}"></span>, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="69" height="18" src="https://math.fontein.de/formulae/3jzUskxusBXQT3tpGWyGvWNb.kOcfS9PxXh8TQ.svgz" alt="\varphi(x) = x" title="\varphi(x) = x"></span>. For any <span class="inline-formula"><img class="img-inline-formula img-formula" width="158" height="18" src="https://math.fontein.de/formulae/1_2zJLGZZfbtkJRIuu8bW0SrTLuQr5D7dXzCDA.svgz" alt="t = (t_1, \dots, t_n) \in \R^n" title="t = (t_1, \dots, t_n) \in \R^n"></span>, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="180" height="18" src="https://math.fontein.de/formulae/LRj4SAENqRQFgkPjCsUYtQgCcl2SUhSpBmxIZg.svgz" alt="x = (x_1, \dots, x_n) = \varphi(t)" title="x = (x_1, \dots, x_n) = \varphi(t)"></span>, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="143" height="23" src="https://math.fontein.de/formulae/4tMdlK9rRFiUR55ShqBOdeWmlJwKh0dWUDzY4Q.svgz" alt="0 \le t_i - x_i \le \frac{A - C}{b_i}" title="0 \le t_i - x_i \le \frac{A - C}{b_i}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="195" height="20" src="https://math.fontein.de/formulae/aPWtfqSLBB11cwISVNzA7Xj6p9rao65BOy9g1Q.svgz" alt="C \le \ell(t) - \sum_{i=1}^n t_i b_i \le A" title="C \le \ell(t) - \sum_{i=1}^n t_i b_i \le A"></span>. In fact, <span class="inline-formula"><img class="img-inline-formula img-formula" width="192" height="20" src="https://math.fontein.de/formulae/IXZtXL6JweedBSbqK9cs_j2YJK.7sait_Re3kQ.svgz" alt="\sum_{i=1}^n (t_i - x_i) b_i \le A - C" title="\sum_{i=1}^n (t_i - x_i) b_i \le A - C"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
For the first statement, it suffices to show <span class="inline-formula"><img class="img-inline-formula img-formula" width="83" height="18" src="https://math.fontein.de/formulae/GFTPN99Qyc88GP_vwLr_uDaS1nGuBwIa6Nq6Cg.svgz" alt="x \in B_{x,\ell(x)}" title="x \in B_{x,\ell(x)}"></span>. But note that if <span class="inline-formula"><img class="img-inline-formula img-formula" width="83" height="19" src="https://math.fontein.de/formulae/UA5eLMC2SbZSLNkOmLzGEUA39uaAsGTfQq3qBg.svgz" alt="x \not\in B_{x,\ell(x)}" title="x \not\in B_{x,\ell(x)}"></span>, we would have <span class="inline-formula"><img class="img-inline-formula img-formula" width="95" height="18" src="https://math.fontein.de/formulae/yv8tblI2svZTmEyh9rRoaj1P0QAaE.79NIDq1A.svgz" alt="\ell(x) > \deg x" title="\ell(x) > \deg x"></span> and hence <span class="inline-formula"><img class="img-inline-formula img-formula" width="98" height="18" src="https://math.fontein.de/formulae/QvcsFHS3IaErKaTFFAdqAqFuN0ZHe.aoLTNjIw.svgz" alt="B_{x,\ell(x)} \subseteq B_x" title="B_{x,\ell(x)} \subseteq B_x"></span>, a contradiction.
</p>
<p>
For the second statement, note that <span class="inline-formula"><img class="img-inline-formula img-formula" width="302" height="20" src="https://math.fontein.de/formulae/a3MSTWaMeitlcWYEh4N5XS_sJSnAb9lrNbLPvg.svgz" alt="\ell(t) = \deg (x_1, \dots, x_n) \le \sum_{i=1}^n x_i b_i + A" title="\ell(t) = \deg (x_1, \dots, x_n) \le \sum_{i=1}^n x_i b_i + A"></span>. Moreover, <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="18" src="https://math.fontein.de/formulae/twc2WH9drdXQbN4Tg6xfQbkaL9hcUD1uEBCx.A.svgz" alt="B_{t,\ell} \neq \emptyset" title="B_{t,\ell} \neq \emptyset"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="138" height="20" src="https://math.fontein.de/formulae/FoSgM0ORoZR2dYnW7UlPWjf9j2_igKkD550szg.svgz" alt="\ell = \sum_{i=1}^n t_i b_i + C" title="\ell = \sum_{i=1}^n t_i b_i + C"></span>, whence we get <span class="inline-formula"><img class="img-inline-formula img-formula" width="282" height="20" src="https://math.fontein.de/formulae/LOPKBYv3DoblbxLWD3gQoHQKudc595UZPO4rFw.svgz" alt="\ell(t) - \sum_{i=1}^n t_i b_i \ge \ell - \sum_{i=1}^n t_i b_i = C" title="\ell(t) - \sum_{i=1}^n t_i b_i \ge \ell - \sum_{i=1}^n t_i b_i = C"></span>. This shows the inequality on <span class="inline-formula"><img class="img-inline-formula img-formula" width="28" height="18" src="https://math.fontein.de/formulae/dFC2dDvk2qsftR7cYnEhXMuVesGebQsygLhB8Q.svgz" alt="\ell(t)" title="\ell(t)"></span>. Now clearly <span class="inline-formula"><img class="img-inline-formula img-formula" width="52" height="14" src="https://math.fontein.de/formulae/Cb6_nW4kgUzAGP_tB1WPFJRDMdAT8ClvJECjIg.svgz" alt="x_i \le t_i" title="x_i \le t_i"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="14" src="https://math.fontein.de/formulae/73Dk8I2n6ASMtmZKrVVByLVQMyS5r6fHYcTe2A.svgz" alt="0 \le t_i - x_i" title="0 \le t_i - x_i"></span>. Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="433" height="20" src="https://math.fontein.de/formulae/m8GK5PHHMNE_6qrXCDDHJxjOM8eIprSm_TYZJA.svgz" alt="A + \sum_{i=1}^n x_i b_i \ge \deg (x_1, \dots, x_n) = \ell(t) \ge C + \sum_{i=1}^n t_i b_i" title="A + \sum_{i=1}^n x_i b_i \ge \deg (x_1, \dots, x_n) = \ell(t) \ge C + \sum_{i=1}^n t_i b_i"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="192" height="20" src="https://math.fontein.de/formulae/IXZtXL6JweedBSbqK9cs_j2YJK.7sait_Re3kQ.svgz" alt="\sum_{i=1}^n (t_i - x_i) b_i \le A - C" title="\sum_{i=1}^n (t_i - x_i) b_i \le A - C"></span>. As <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="14" src="https://math.fontein.de/formulae/_LPmWTXOe7Qa.8xJEh369zlB4a5TfbIglW.5bA.svgz" alt="t_i - x_i \ge 0" title="t_i - x_i \ge 0"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
</div>
</div>
<div class="subsection-block">
<h4>
Using Minima of Lattices.
</h4>
<div>
<p>
In this section, we describe how to obtain <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="17" src="https://math.fontein.de/formulae/trC8ROiBHCvA1Mxjz0DMydgaaneA8jHc1Qwtww.svgz" alt="\hat{X}" title="\hat{X}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="11" src="https://math.fontein.de/formulae/G0SX86eTASn_9M49WF7HzDK85n7NoD6NrqM3ew.svgz" alt="\varphi" title="\varphi"></span> from an <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="18" src="https://math.fontein.de/formulae/AjCyaw2HagCBfGhDL8_wwlnYMLlETAC11TNcKw.svgz" alt="(n + 1)" title="(n + 1)"></span>-dimensional lattice <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="17" src="https://math.fontein.de/formulae/umj9Mm8phHATJvpYo21CHXGOnkjX.rspFd4KHw.svgz" alt="\Gamma \subseteq \R^{n+1}" title="\Gamma \subseteq \R^{n+1}"></span>. We require that for every <span class="inline-formula"><img class="img-inline-formula img-formula" width="164" height="18" src="https://math.fontein.de/formulae/VbdNX9UoGPMSZu_Xd49M_Iw5989CgniXIcuQMQ.svgz" alt="t = (t_1, \dots, t_{n+1}) \in \Gamma" title="t = (t_1, \dots, t_{n+1}) \in \Gamma"></span>, we either have <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="11" src="https://math.fontein.de/formulae/Ni_G35Kd.jrxuXXxFr1Yd09FfTxPurGcQO2uVg.svgz" alt="t = 0" title="t = 0"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="16" src="https://math.fontein.de/formulae/4jBePBR_iKNMcT..9hGF6u00vBDtqyG2fWonEg.svgz" alt="t_i \neq 0" title="t_i \neq 0"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span>. More precisely, consider the map <span class="inline-formula"><img class="img-inline-formula img-formula" width="112" height="14" src="https://math.fontein.de/formulae/3h0gAW.51OAmb3_N7Mi3sKmHjvLvJ38zB8QJiQ.svgz" alt="N : \R^{n+1} \to \R" title="N : \R^{n+1} \to \R"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="196" height="20" src="https://math.fontein.de/formulae/uiwJGvqG41uWjou5XlFGhnn9d1yu6X00Km.IZQ.svgz" alt="(x_1, \dots, x_{n+1}) \mapsto \prod_{i=1}^n x_i" title="(x_1, \dots, x_{n+1}) \mapsto \prod_{i=1}^n x_i"></span>. We assume that there exists a constant <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="12" src="https://math.fontein.de/formulae/Y05P0JkvX29VJH4AZy9arj_n37uSaUoGJGq1LA.svgz" alt="c > 0" title="c > 0"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="71" height="18" src="https://math.fontein.de/formulae/YbdsnSSHUsvJEjZIzhcxqeKv3Jw7F0eujjZGxw.svgz" alt="N(x) \ge c" title="N(x) \ge c"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="86" height="18" src="https://math.fontein.de/formulae/WB62vb61nYQ9ktOvqZKFvVPA9Ry3yLtXneHvOA.svgz" alt="x \in \Gamma \setminus \{ 0 \}" title="x \in \Gamma \setminus \{ 0 \}"></span>.
</p>
<p>
In fact, one can replace <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/lZFl0.wtx.TkJAn.ie0DICOmVjwKyUYdr01UZA.svgz" alt="\Gamma" title="\Gamma"></span> by any discrete subset with some additional properties which give similar results as <a href="https://en.wikipedia.org/wiki/Minkowski%2527s_theorem">Minkowski's Lattice Point Theorem</a>.
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<p>
A <em>minimum</em> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/lZFl0.wtx.TkJAn.ie0DICOmVjwKyUYdr01UZA.svgz" alt="\Gamma" title="\Gamma"></span> is an element <span class="inline-formula"><img class="img-inline-formula img-formula" width="220" height="18" src="https://math.fontein.de/formulae/EXl19wcmIC6OUqyWLKs1Dk9i5bwT_ZSPGBut7g.svgz" alt="\mu = (\mu_1, \dots, \mu_{n+1}) \in \Gamma \setminus \{ 0 \}" title="\mu = (\mu_1, \dots, \mu_{n+1}) \in \Gamma \setminus \{ 0 \}"></span> such that for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="170" height="18" src="https://math.fontein.de/formulae/z5e68rwzWYkCyq8LR4XbVtWDaNUt9331ZS6YGA.svgz" alt="z = (z_1, \dots, z_{n+1}) \in \Gamma" title="z = (z_1, \dots, z_{n+1}) \in \Gamma"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="18" src="https://math.fontein.de/formulae/jP7KeOeQZMrluLDL2._53AIHRclCAoYYZgW1ng.svgz" alt="\abs{z_i} \le \abs{\mu_i}" title="\abs{z_i} \le \abs{\mu_i}"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span>, we either have <span class="inline-formula"><img class="img-inline-formula img-formula" width="41" height="11" src="https://math.fontein.de/formulae/qdlaQM2zfjRWP7gPnLf_Nz5Eak8icXb33yVmUQ.svgz" alt="z = 0" title="z = 0"></span> or <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="18" src="https://math.fontein.de/formulae/l0r9YOMauqdKgRcB.JWoVA4yPoE4fUvm8MNlkQ.svgz" alt="\abs{z_i} = \abs{\mu_i}" title="\abs{z_i} = \abs{\mu_i}"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span>. Denote the set of all minima by <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="12" src="https://math.fontein.de/formulae/ru_bzFbUFrZqnMJWjp0utN116NL3qRL_x_19NQ.svgz" alt="\min \Gamma" title="\min \Gamma"></span>.
</p>
</div>
</div>
<p>
First, we will show that such minima exist:
</p>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="207" height="18" src="https://math.fontein.de/formulae/jRZ7S0jR0KOl1M7lyc6nYQs2rEQMMcQt7BV8vA.svgz" alt="t = (t_1, \dots, t_{n+1}) \in \Gamma \setminus \{ 0 \}" title="t = (t_1, \dots, t_{n+1}) \in \Gamma \setminus \{ 0 \}"></span>. Then there exists a minimum <span class="inline-formula"><img class="img-inline-formula img-formula" width="144" height="18" src="https://math.fontein.de/formulae/8AsHieQWUPDcvrN1RyXSe7JC8RCyeW62tOyV_w.svgz" alt="\mu = (\mu_1, \dots, \mu_{n+1})" title="\mu = (\mu_1, \dots, \mu_{n+1})"></span> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/lZFl0.wtx.TkJAn.ie0DICOmVjwKyUYdr01UZA.svgz" alt="\Gamma" title="\Gamma"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="72" height="18" src="https://math.fontein.de/formulae/jZ_dqti_s_RnhPgi4YbyrPZ8jtkGfUZ0VPpyDg.svgz" alt="\abs{\mu_i} \le \abs{t_i}" title="\abs{\mu_i} \le \abs{t_i}"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
This follows from the fact that <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/lZFl0.wtx.TkJAn.ie0DICOmVjwKyUYdr01UZA.svgz" alt="\Gamma" title="\Gamma"></span> is discrete. For <span class="inline-formula"><img class="img-inline-formula img-formula" width="137" height="18" src="https://math.fontein.de/formulae/4yA0u_q9lA9ZQqkaBz_plBM7s4o9C6bQuvGncw.svgz" alt="s = (s_1, \dots, s_{n+1})" title="s = (s_1, \dots, s_{n+1})"></span>, define
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="402" height="18" src="https://math.fontein.de/formulae/ohyAD4pJYYfss7onrNgPBVmp9AFHvZaPI0IvPw.svgz" alt="B_s := \{ (x_1, \dots, x_{n+1} \in \Gamma \setminus \{ 0 \} \mid \abs{x_i} \le \abs{s_i} \text{ for all } i \}." title="B_s := \{ (x_1, \dots, x_{n+1} \in \Gamma \setminus \{ 0 \} \mid \abs{x_i} \le \abs{s_i} \text{ for all } i \}.">
</div>
<p>
As <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/lZFl0.wtx.TkJAn.ie0DICOmVjwKyUYdr01UZA.svgz" alt="\Gamma" title="\Gamma"></span> is discrete, <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="15" src="https://math.fontein.de/formulae/enqXIoYpA1mJdGnk0TcrmbZRmm0Sl.WXRKnlOg.svgz" alt="B_s" title="B_s"></span> is always finite.
</p>
<p>
In particular, <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/YnedDTdUNhPbVh1L471N4I7CWnAUjq3Zead0UA.svgz" alt="B_t" title="B_t"></span> is finite. Assume that <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="11" src="https://math.fontein.de/formulae/Fb4QjIncBFMP2NDS0yCZ5Fvi7l8GB681giwdQQ.svgz" alt="t" title="t"></span> is not a minimum (in which case we could choose <span class="inline-formula"><img class="img-inline-formula img-formula" width="41" height="14" src="https://math.fontein.de/formulae/G5TRZd96c.wLPOzOsBYfOeT30o08EY6iWmkqtA.svgz" alt="\mu = t" title="\mu = t"></span>). Then there exists some <span class="inline-formula"><img class="img-inline-formula img-formula" width="178" height="18" src="https://math.fontein.de/formulae/0LF5C17o70VcBDkuRJvnah0GHAeITtAz8jiBRA.svgz" alt="s = (s_1, \dots, s_{n+1}) \in B_t" title="s = (s_1, \dots, s_{n+1}) \in B_t"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="70" height="18" src="https://math.fontein.de/formulae/BGKfVTugGuxNvirtqstacAr5bhNFcXQlv.atHw.svgz" alt="\abs{s_i} < \abs{t_i}" title="\abs{s_i} < \abs{t_i}"></span> for some <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span>. In that case, <span class="inline-formula"><img class="img-inline-formula img-formula" width="94" height="19" src="https://math.fontein.de/formulae/LJsMABvDaUHMAclz5cxZCjc.rqp.eh13PWgdjw.svgz" alt="s \in B_s \subsetneqq B_t" title="s \in B_s \subsetneqq B_t"></span>. Now either <span class="inline-formula"><img class="img-inline-formula img-formula" width="8" height="8" src="https://math.fontein.de/formulae/ovmFNULo3EfWiVyyUOID62AW15.M_94ZehNNBw.svgz" alt="s" title="s"></span> is a minimum, in which case we choose <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="11" src="https://math.fontein.de/formulae/qfebFXv8bOrowFpmDzQ9s7RozYyjR8vxSw5lhQ.svgz" alt="\mu = s" title="\mu = s"></span>, or it is not. In that case, we can repeat the procedure with <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="15" src="https://math.fontein.de/formulae/enqXIoYpA1mJdGnk0TcrmbZRmm0Sl.WXRKnlOg.svgz" alt="B_s" title="B_s"></span> instead of <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/YnedDTdUNhPbVh1L471N4I7CWnAUjq3Zead0UA.svgz" alt="B_t" title="B_t"></span>. As the size of these sets decreases every step and the sets are finite but non-empty, we eventually must find some <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="15" src="https://math.fontein.de/formulae/ZzRpV_wYBiM4pdHeO5khRf9qtzy1JBpXPuMxvA.svgz" alt="s \in B_t" title="s \in B_t"></span> which is a minimum.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Define <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="6" src="https://math.fontein.de/formulae/7CCy3eLHRu9gOobKOrqS1H_e_JbMkBDj7nC25Q.svgz" alt="\sim" title="\sim"></span> on <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="14" src="https://math.fontein.de/formulae/yo3D3yFnt1B05mNyv8ERQPdEDUGS70ry2cm.kQ.svgz" alt="\R^{n+1}" title="\R^{n+1}"></span> by
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="383" height="18" src="https://math.fontein.de/formulae/qCYy7_DbIkFKrm92FogOxPnQGH51YIpZIxDNYQ.svgz" alt="(s_1, \dots, s_{n+1}) \sim (t_1, \dots, t_{n+1}) :\Longleftrightarrow \forall i : \abs{s_i} = \abs{t_i}," title="(s_1, \dots, s_{n+1}) \sim (t_1, \dots, t_{n+1}) :\Longleftrightarrow \forall i : \abs{s_i} = \abs{t_i},">
</div>
<p>
and consider the map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="442" height="18" src="https://math.fontein.de/formulae/.Cu1IS6H.4qDOrIT6j00UIvvM6oSlecPc_8wUA.svgz" alt="\Phi : \Gamma \setminus \{ 0 \} \to \R^n, \quad (t_1, \dots, t_{n+1}) = (\log \abs{t_1}, \dots, \log \abs{t_n})." title="\Phi : \Gamma \setminus \{ 0 \} \to \R^n, \quad (t_1, \dots, t_{n+1}) = (\log \abs{t_1}, \dots, \log \abs{t_n}).">
</div>
<p>
First, <span class="inline-formula"><img class="img-inline-formula img-formula" width="94" height="18" src="https://math.fontein.de/formulae/mhgk7R9MFh6qlrEk9r8CKs8znk72i7xShezRIg.svgz" alt="\Phi(a) = \Phi(b)" title="\Phi(a) = \Phi(b)"></span> if, and only if, <span class="inline-formula"><img class="img-inline-formula img-formula" width="41" height="12" src="https://math.fontein.de/formulae/H0Z2Sio9TM5Z_h1q1u9XdogLe97mpf8MX1FG_g.svgz" alt="a \sim b" title="a \sim b"></span>. Let
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="340" height="21" src="https://math.fontein.de/formulae/QsEDigz5MBunBvZ6Q2rtmFGQIfJ5cjLmZbox2A.svgz" alt="\hat{X} := \Phi(\min \Gamma) = \{ \Phi(\mu) \mid \mu \text{ minimum of } \Gamma \}." title="\hat{X} := \Phi(\min \Gamma) = \{ \Phi(\mu) \mid \mu \text{ minimum of } \Gamma \}.">
</div>
<div class="theorem-environment theorem-lemma-environment">
<div class="theorem-header theorem-lemma-header">
Lemma.
</div>
<div class="theorem-content theorem-lemma-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="158" height="18" src="https://math.fontein.de/formulae/1_2zJLGZZfbtkJRIuu8bW0SrTLuQr5D7dXzCDA.svgz" alt="t = (t_1, \dots, t_n) \in \R^n" title="t = (t_1, \dots, t_n) \in \R^n"></span>. Then, there exists some <span class="inline-formula"><img class="img-inline-formula img-formula" width="164" height="21" src="https://math.fontein.de/formulae/xWuXvtVRjZq_K5lYgeNfOorZtjvzJk7_KlBuYw.svgz" alt="\mu = (\mu_1, \dots, \mu_n) \in \hat{X}" title="\mu = (\mu_1, \dots, \mu_n) \in \hat{X}"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="14" src="https://math.fontein.de/formulae/73Dk8I2n6ASMtmZKrVVByLVQMyS5r6fHYcTe2A.svgz" alt="0 \le t_i - x_i" title="0 \le t_i - x_i"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="204" height="20" src="https://math.fontein.de/formulae/yKVRcQT_NuTUKH0xsggxiJbV.5SjjlUz3pm98w.svgz" alt="\sum_{i=1}^n (t_i - x_i) \le \log \abs{\det \Gamma}" title="\sum_{i=1}^n (t_i - x_i) \le \log \abs{\det \Gamma}"></span>. In particular, <span class="inline-formula"><img class="img-inline-formula img-formula" width="148" height="18" src="https://math.fontein.de/formulae/IZB6BnV8IsMK23nZBbtgwORqKHYRW2._g_XUAg.svgz" alt="t_i - x_i \le \log \abs{\det \Gamma}" title="t_i - x_i \le \log \abs{\det \Gamma}"></span>.
</p>
</div>
</div>
<p>
Here, <span class="inline-formula"><img class="img-inline-formula img-formula" width="39" height="12" src="https://math.fontein.de/formulae/v1G05ZGNcOYUnpjbRFljDWYop5dzWZLhHhdMcA.svgz" alt="\det{\Gamma}" title="\det{\Gamma}"></span> is the determinant of the lattice <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/lZFl0.wtx.TkJAn.ie0DICOmVjwKyUYdr01UZA.svgz" alt="\Gamma" title="\Gamma"></span>, i.e. the volume of one fundamental parallelepiped of <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/lZFl0.wtx.TkJAn.ie0DICOmVjwKyUYdr01UZA.svgz" alt="\Gamma" title="\Gamma"></span>.
</p>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
For <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="13" src="https://math.fontein.de/formulae/sajLq_SKey_MIohLJPM5_HqE8PXy.kXIJ5uy2A.svgz" alt="\ell > 0" title="\ell > 0"></span>, consider the set
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="451" height="20" src="https://math.fontein.de/formulae/RmrTrZm4M9NMrMCl5343.weFiGQN2eAACd7p1w.svgz" alt="B_\ell := \{ (x_1, \dots, x_{n+1}) \in \R^{n+1} \mid \abs{x_i} \le \exp(t_i), \; \abs{x_{n+1}} \le \ell \}." title="B_\ell := \{ (x_1, \dots, x_{n+1}) \in \R^{n+1} \mid \abs{x_i} \le \exp(t_i), \; \abs{x_{n+1}} \le \ell \}.">
</div>
<p>
By Minkowski's Lattice Point Theorem, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="83" height="16" src="https://math.fontein.de/formulae/WaTtkDud9gWUX8J06o28kJr2JomaExxGaAwcEQ.svgz" alt="B_\ell \cap \Gamma \neq 0" title="B_\ell \cap \Gamma \neq 0"></span> for
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="323" height="52" src="https://math.fontein.de/formulae/suCFMIc_hiTW0LkoGlMNV4VdzLV54NHSTB2rZw.svgz" alt="2^n \prod_{i=1}^n \exp(t_i) \cdot 2 \ell = \mathrm{vol}(B_\ell) > 2^{n+1} \abs{\det \Gamma}," title="2^n \prod_{i=1}^n \exp(t_i) \cdot 2 \ell = \mathrm{vol}(B_\ell) > 2^{n+1} \abs{\det \Gamma},">
</div>
<p>
i.e.
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="198" height="52" src="https://math.fontein.de/formulae/RwpOT8zdY4iDUx1xOH3tWgnCnCbIpUBVuWijig.svgz" alt="\ell > \abs{\det \Gamma} \exp\biggl( -\sum_{i=1}^n t_i \biggr)." title="\ell > \abs{\det \Gamma} \exp\biggl( -\sum_{i=1}^n t_i \biggr).">
</div>
<p>
Since <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/j0a55wkBPqZKflQEb.noEqsh0kkwQvybOzqNfg.svgz" alt="B_\ell" title="B_\ell"></span> is closed and <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/lZFl0.wtx.TkJAn.ie0DICOmVjwKyUYdr01UZA.svgz" alt="\Gamma" title="\Gamma"></span> discrete, a limit argument shows that this also holds for <span class="inline-formula"><img class="img-inline-formula img-formula" width="200" height="21" src="https://math.fontein.de/formulae/r6clnBU4U3ADap_PYCB8VB9kE9ozVD08mDaU_w.svgz" alt="\ell = \abs{\det \Gamma} \exp\bigl( -\sum_{i=1}^n t_i \bigr)" title="\ell = \abs{\det \Gamma} \exp\bigl( -\sum_{i=1}^n t_i \bigr)"></span>. By the previous lemma, there exists a minimum <span class="inline-formula"><img class="img-inline-formula img-formula" width="137" height="18" src="https://math.fontein.de/formulae/4yA0u_q9lA9ZQqkaBz_plBM7s4o9C6bQuvGncw.svgz" alt="s = (s_1, \dots, s_{n+1})" title="s = (s_1, \dots, s_{n+1})"></span> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/lZFl0.wtx.TkJAn.ie0DICOmVjwKyUYdr01UZA.svgz" alt="\Gamma" title="\Gamma"></span> which lies in <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="15" src="https://math.fontein.de/formulae/j0a55wkBPqZKflQEb.noEqsh0kkwQvybOzqNfg.svgz" alt="B_\ell" title="B_\ell"></span>; let <span class="inline-formula"><img class="img-inline-formula img-formula" width="195" height="18" src="https://math.fontein.de/formulae/eI_3Rtky52QtT6QpJwHNmWC.d6_2ZEOavVsCEQ.svgz" alt="\mu := (\mu_1, \dots, \mu_n) := \Phi(s)" title="\mu := (\mu_1, \dots, \mu_n) := \Phi(s)"></span>. Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="229" height="18" src="https://math.fontein.de/formulae/Cc7OgvLw2fD_3eBCpCkP2NUP8g774LIFKn2dmA.svgz" alt="\mu_i = \log \abs{s_i} \le \log \exp(t_i) = t_i" title="\mu_i = \log \abs{s_i} \le \log \exp(t_i) = t_i"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="73" height="14" src="https://math.fontein.de/formulae/FpZ4n1WCX1xSEeoMKKG9Jn.6P1n9K4u5Fncztw.svgz" alt="1 \le i \le n" title="1 \le i \le n"></span> as <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="15" src="https://math.fontein.de/formulae/KI2yUvpeXkJXSZaXP7vgHVSQqp5TX8OULUfOvQ.svgz" alt="s \in B_\ell" title="s \in B_\ell"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="83" height="15" src="https://math.fontein.de/formulae/DJvfPzQZe10AtOv.Gfzms7boHxgduuWrJHQ_5g.svgz" alt="0 \le t_i - \mu_i" title="0 \le t_i - \mu_i"></span>.
</p>
<p>
Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="70" height="18" src="https://math.fontein.de/formulae/yXRZsYrLB_RX27PXQCDjs6EiblhfUStmvbUCBQ.svgz" alt="N(s) \ge c" title="N(s) \ge c"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="212" height="20" src="https://math.fontein.de/formulae/MBv5XJbLbzQazAA.h2Gim6BTZhF3LszoO.1CgQ.svgz" alt="\sum_{i=1}^n \mu_i \ge \log c - \log \abs{s_{n+1}}" title="\sum_{i=1}^n \mu_i \ge \log c - \log \abs{s_{n+1}}"></span>. Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="77" height="18" src="https://math.fontein.de/formulae/kVd4tJkSy2JVGQSO98wIqMaE3EUSn2k_o_JRFA.svgz" alt="\abs{s_{n+1}} \le \ell" title="\abs{s_{n+1}} \le \ell"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="356" height="20" src="https://math.fontein.de/formulae/91wOArDXU_GCeaVcVCBWQbzvjvGb20cxi9tWkA.svgz" alt="-\log \abs{s_{n+1}} \ge -\log \ell \ge -\log \abs{\det \Gamma} + \sum_{i=1}^n t_i" title="-\log \abs{s_{n+1}} \ge -\log \ell \ge -\log \abs{\det \Gamma} + \sum_{i=1}^n t_i"></span>. Therefore, we get
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="230" height="52" src="https://math.fontein.de/formulae/6jceqs6S_54tAHoCi1AVd7gakmVAHWEqgt4e0A.svgz" alt="\sum_{i=1}^n \mu_i \ge -\log \abs{\det \Gamma} + \sum_{i=1}^n t_i," title="\sum_{i=1}^n \mu_i \ge -\log \abs{\det \Gamma} + \sum_{i=1}^n t_i,">
</div>
<p>
i.e. <span class="inline-formula"><img class="img-inline-formula img-formula" width="205" height="20" src="https://math.fontein.de/formulae/.DMnDYZu5usGKfncjmJeUcMbd3MQIPFOk1mmIQ.svgz" alt="\sum_{i=1}^n (t_i - \mu_i) \le \log \abs{\det \Gamma}" title="\sum_{i=1}^n (t_i - \mu_i) \le \log \abs{\det \Gamma}"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Define <span class="inline-formula"><img class="img-inline-formula img-formula" width="281" height="21" src="https://math.fontein.de/formulae/g5mp47v45GodX8riOLk0LgZiyeiyxy6n2YL3WA.svgz" alt="\Lambda := \{ x \in \R^n \mid \forall \mu \in \hat{X} : x + \mu \in \hat{X} \}" title="\Lambda := \{ x \in \R^n \mid \forall \mu \in \hat{X} : x + \mu \in \hat{X} \}"></span>; this is a discrete subgroup of <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/bZVXuWhECmRV9pmTW_AgHZeLTOYax7svHPDCxQ.svgz" alt="\R^n" title="\R^n"></span>. <strong>We assume that <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/XY_2LK590TGO0.jb0ZbkmaiBi4kJr1xvp52q2g.svgz" alt="\Lambda" title="\Lambda"></span> is a lattice in <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/bZVXuWhECmRV9pmTW_AgHZeLTOYax7svHPDCxQ.svgz" alt="\R^n" title="\R^n"></span></strong>, i.e. contains a basis of <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/bZVXuWhECmRV9pmTW_AgHZeLTOYax7svHPDCxQ.svgz" alt="\R^n" title="\R^n"></span>. We can define <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="111" height="18" src="https://math.fontein.de/formulae/2pzYFU99Q.eG3U9ebfYQlDgAMSYK7GaQKzO8cg.svgz" alt="d : X \to \R^n/\Lambda" title="d : X \to \R^n/\Lambda"></span> such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="122" height="21" src="https://math.fontein.de/formulae/n8Ey1Qii.IRtWcxyvn1Cz6PC2yHSWHc4bnpNWQ.svgz" alt="\hat{X} = \pi^{-1}(d(X))" title="\hat{X} = \pi^{-1}(d(X))"></span>, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="119" height="18" src="https://math.fontein.de/formulae/XHL69cR1ss78J_iGymmCG4VcBZ941OC3CarIdg.svgz" alt="\pi : \R^n \to \R^n/\Lambda" title="\pi : \R^n \to \R^n/\Lambda"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="14" src="https://math.fontein.de/formulae/45iAV4bRHY0nr_zh4opD7.I5mxgs4r32sxsAgg.svgz" alt="t \mapsto t + \Lambda" title="t \mapsto t + \Lambda"></span> is the projection. To get an <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructure, we are left to define <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="11" src="https://math.fontein.de/formulae/G0SX86eTASn_9M49WF7HzDK85n7NoD6NrqM3ew.svgz" alt="\varphi" title="\varphi"></span>.
</p>
<p>
For that, we proceed as in the proof of the second lemma in this section. For <span class="inline-formula"><img class="img-inline-formula img-formula" width="158" height="18" src="https://math.fontein.de/formulae/1_2zJLGZZfbtkJRIuu8bW0SrTLuQr5D7dXzCDA.svgz" alt="t = (t_1, \dots, t_n) \in \R^n" title="t = (t_1, \dots, t_n) \in \R^n"></span>, consider
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="351" height="43" src="https://math.fontein.de/formulae/YqwBLnzLd4OZMkTp22sOlSAj7U.OIQXCGf9w7A.svgz" alt="B_\ell := \biggl\{ \Psi(x) \;\biggm| \begin{matrix} x = (x_1, \dots, x_{n+1}) \in \min \Gamma, \\ \abs{x_i} \le \exp(t_i), \; \abs{x_{n+1}} \le \ell \end{matrix} \biggr\}." title="B_\ell := \biggl\{ \Psi(x) \;\biggm| \begin{matrix} x = (x_1, \dots, x_{n+1}) \in \min \Gamma, \\ \abs{x_i} \le \exp(t_i), \; \abs{x_{n+1}} \le \ell \end{matrix} \biggr\}.">
</div>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="13" src="https://math.fontein.de/formulae/sajLq_SKey_MIohLJPM5_HqE8PXy.kXIJ5uy2A.svgz" alt="\ell > 0" title="\ell > 0"></span> be minimal with <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="17" src="https://math.fontein.de/formulae/EkOqccQu9awmchrbLe6gwvbBrL1TN5x.KW1v3w.svgz" alt="B_\ell \neq \emptyset" title="B_\ell \neq \emptyset"></span>, and let <span class="inline-formula"><img class="img-inline-formula img-formula" width="128" height="18" src="https://math.fontein.de/formulae/2e6ufUfbbtfJCRpDTbgHfY69sMHi78_KZ00b5w.svgz" alt="\varphi(t) := \max_{\le} B_\ell" title="\varphi(t) := \max_{\le} B_\ell"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="32" height="18" src="https://math.fontein.de/formulae/k1RD0iQX4eTrtI8VjjOpzF5Ybp4cta7_h0YSsQ.svgz" alt="\varphi(t)" title="\varphi(t)"></span> satisfies the properties in the statement of the lemma, i.e. lies near to <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="11" src="https://math.fontein.de/formulae/Fb4QjIncBFMP2NDS0yCZ5Fvi7l8GB681giwdQQ.svgz" alt="t" title="t"></span> itself. Moreover, one quickly checks that <span class="inline-formula"><img class="img-inline-formula img-formula" width="151" height="18" src="https://math.fontein.de/formulae/Sp.xXybT5sfo8dY_rtA4e2wZPQC.DLsZ67B.wA.svgz" alt="\varphi(t + \lambda) = \varphi(t) + \lambda" title="\varphi(t + \lambda) = \varphi(t) + \lambda"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="13" src="https://math.fontein.de/formulae/A5CppZBos4lcq9WsAE.53BBJujPwtmvA8McOnA.svgz" alt="\lambda \in \Lambda" title="\lambda \in \Lambda"></span>.
</p>
<p>
Hence, we obtain an <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructure.
</p>
</div>
</div>
</div>
n-dimensional Infrastructures.
https://math.fontein.de/2009/07/20/n-dimensional-infrastructures/
2009-07-20T08:40:46+02:00
2009-07-20T08:40:46+02:00
Felix Fontein
<div>
<div>
<p>
For <a href="https://math.fontein.de/2009/07/20/one-dimensional-infrastructures/">one-dimensional infrastructures</a>, we have a circle <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="18" src="https://math.fontein.de/formulae/Blo1FniNoktf2EKQy0CWZxv_r_wmqWWcM_Pc9A.svgz" alt="\R/R\Z" title="\R/R\Z"></span> together with a finite, non-empty set <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> and an injective map <span class="inline-formula"><img class="img-inline-formula img-formula" width="115" height="18" src="https://math.fontein.de/formulae/nG5tOQuC_AiH8MK1kd7kHlmnr7XkfyU3Es3Tew.svgz" alt="d : X \to \R/R\Z" title="d : X \to \R/R\Z"></span>. Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="114" height="18" src="https://math.fontein.de/formulae/je5sWDSzyU2LwOWbO2BPl7MrRpZMNNPO4l_UJg.svgz" alt="\R/R\Z = \R^n / \Lambda" title="\R/R\Z = \R^n / \Lambda"></span>, where <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="11" src="https://math.fontein.de/formulae/AI3jLOaiR4OxpS_JgU78KWnQ5jO4jP6BS3hJew.svgz" alt="n = 1" title="n = 1"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/XY_2LK590TGO0.jb0ZbkmaiBi4kJr1xvp52q2g.svgz" alt="\Lambda" title="\Lambda"></span> is the one-dimensional <a href="https://en.wikipedia.org/wiki/Lattice_%28group%29">lattice</a> <span class="inline-formula"><img class="img-inline-formula img-formula" width="61" height="12" src="https://math.fontein.de/formulae/ehoSwEBlGxIA1X6WzvZdPs3rJo291gV1954Ubw.svgz" alt="\Lambda = R \Z" title="\Lambda = R \Z"></span>. Hence, one could say that an <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructure is a torus <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="18" src="https://math.fontein.de/formulae/W.nVj7Zi0tCSZE0Q2eogNxSUumdXLxoOTJEzCw.svgz" alt="\R^n/\Lambda" title="\R^n/\Lambda"></span> together with <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="111" height="18" src="https://math.fontein.de/formulae/2pzYFU99Q.eG3U9ebfYQlDgAMSYK7GaQKzO8cg.svgz" alt="d : X \to \R^n/\Lambda" title="d : X \to \R^n/\Lambda"></span> as above. From the discussion in the remarks of <a href="https://math.fontein.de/2009/07/20/one-dimensional-infrastructures/">this post</a> we see that we need some kind of reduction map to define giant steps (and also <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-representations) in the one-dimensional case, even though there a pretty canonical reduction map is given. In the case of <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="18" src="https://math.fontein.de/formulae/W.nVj7Zi0tCSZE0Q2eogNxSUumdXLxoOTJEzCw.svgz" alt="\R^n/\Lambda" title="\R^n/\Lambda"></span>, we do not have something similar to a given positive direction. Moreover, definiting the “nearest” element of a finite subset of <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="18" src="https://math.fontein.de/formulae/W.nVj7Zi0tCSZE0Q2eogNxSUumdXLxoOTJEzCw.svgz" alt="\R^n/\Lambda" title="\R^n/\Lambda"></span> to some <span class="inline-formula"><img class="img-inline-formula img-formula" width="72" height="18" src="https://math.fontein.de/formulae/dSbvCoREOPLKRZwVZS3oVPS8tuJgehNh0C1G7w.svgz" alt="t \in \R^n/\Lambda" title="t \in \R^n/\Lambda"></span> is even more complicated and offers more choices which appear more or less obvious. Only the selection of different norms on <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/bZVXuWhECmRV9pmTW_AgHZeLTOYax7svHPDCxQ.svgz" alt="\R^n" title="\R^n"></span> lead to several possible definitions of such a map. Hence, we should require such a map in the definition:
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="15" src="https://math.fontein.de/formulae/EBTY1cww2.XbyZp0zeo5cPIgVso4OVcWdqrNBg.svgz" alt="\Lambda \subseteq \R^n" title="\Lambda \subseteq \R^n"></span> be a lattice. Then, an <em><span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructure</em> is a non-empty finite set <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> together with an injective map <span class="inline-formula"><img class="img-inline-formula img-formula" width="111" height="18" src="https://math.fontein.de/formulae/g3EkzKjEux19bH4SPfjRnoZOQOaLVxwmoBd9LA.svgz" alt="d : X \to \R^n / \Lambda" title="d : X \to \R^n / \Lambda"></span> and another map <span class="inline-formula"><img class="img-inline-formula img-formula" width="128" height="18" src="https://math.fontein.de/formulae/HrdulxFFRS4ZXBKuq8UY4ZHuLaDvvXLDps7nhA.svgz" alt="red : \R^n/\Lambda \to X" title="red : \R^n/\Lambda \to X"></span> satisfying <span class="inline-formula"><img class="img-inline-formula img-formula" width="104" height="15" src="https://math.fontein.de/formulae/3TNT2dOT2EXA3eU7nQ52xNGV4zlFqL_Gje1h0Q.svgz" alt="red \circ d = \id_X" title="red \circ d = \id_X"></span>.
</p>
</div>
</div>
<p>
Again, as in the one-dimensional case, one can define giant steps:
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="327" height="19" src="https://math.fontein.de/formulae/xcADjETDmD0MW7QB7UvF10FS_sfkTdN3JEhDcA.svgz" alt="\gs(x, x') := red(d(x) + d(x')), \quad x, x' \in X." title="\gs(x, x') := red(d(x) + d(x')), \quad x, x' \in X.">
</div>
<p>
Moreover, one gets the same relation between reduction maps and <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>-representations, whence we define
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="518" height="20" src="https://math.fontein.de/formulae/QJN0BpLAOPpkagUbNlY5LvYA.o626.T2VVV4Gg.svgz" alt="\fRep := \fRep(X, d, red) := \{ (x, f) \in X \times \R^n \mid red(d(x) + f) = x \}." title="\fRep := \fRep(X, d, red) := \{ (x, f) \in X \times \R^n \mid red(d(x) + f) = x \}.">
</div>
<p>
Then the map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="379" height="20" src="https://math.fontein.de/formulae/D49Sf0cS7t_ihscO0AypGfxiMGwMb7s_UPr.HQ.svgz" alt="\Psi : \fRep(X, d, red) \to \R^n/\Lambda, \quad (x, f) \mapsto d(x) + f" title="\Psi : \fRep(X, d, red) \to \R^n/\Lambda, \quad (x, f) \mapsto d(x) + f">
</div>
<p>
is a bijection, and we can use this bijection to equip <span class="inline-formula"><img class="img-inline-formula img-formula" width="119" height="19" src="https://math.fontein.de/formulae/R1_fB035Gv00zHSrkUbMpoa1r9AqRh2GRWmwZA.svgz" alt="\fRep(X, d, red)" title="\fRep(X, d, red)"></span> with a group law by
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="527" height="20" src="https://math.fontein.de/formulae/Qb8TPwpT4rhUfYu0KDJMlkhUtHka.2Xqjfko4g.svgz" alt="(x, f) + (x', f') = \Psi^{-1}(\Psi(x, f) + \Psi(x', f')), \quad (x, f), (x', f') \in \fRep." title="(x, f) + (x', f') = \Psi^{-1}(\Psi(x, f) + \Psi(x', f')), \quad (x, f), (x', f') \in \fRep.">
</div>
</div>
<div class="subsection-block">
<h4>
Discrete Infrastructure.
</h4>
<div>
<p>
We say that an <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructure <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="18" src="https://math.fontein.de/formulae/Q3Wi7WBtdKSnv1gaLvYnlftQgM.JrmfkFL4zUA.svgz" alt="(X, d, red)" title="(X, d, red)"></span> with lattice <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/XY_2LK590TGO0.jb0ZbkmaiBi4kJr1xvp52q2g.svgz" alt="\Lambda" title="\Lambda"></span> is <em>discrete</em> if <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="15" src="https://math.fontein.de/formulae/J8WZLQNTj4hsLiLptic9ZbHN6du478Qep2spzQ.svgz" alt="\Lambda \subseteq \Z^n" title="\Lambda \subseteq \Z^n"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="105" height="18" src="https://math.fontein.de/formulae/dulT4cQCEDJN0_jeueOyj_j1g565IBQsowZkQA.svgz" alt="d(X) \subseteq \Z^n/\Lambda" title="d(X) \subseteq \Z^n/\Lambda"></span> and if <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> does not depends on fractions. To make the last part more precise, define
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="400" height="18" src="https://math.fontein.de/formulae/JKoXxyyo.1bxGK5IEGxnKqkfKl3j4oaKZqWa_g.svgz" alt="floor : \R^n \to \Z^n, \quad (x_1, \dots, x_n) \mapsto (\floor{x_1}, \dots, \floor{x_n});" title="floor : \R^n \to \Z^n, \quad (x_1, \dots, x_n) \mapsto (\floor{x_1}, \dots, \floor{x_n});">
</div>
<p>
if <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="15" src="https://math.fontein.de/formulae/J8WZLQNTj4hsLiLptic9ZbHN6du478Qep2spzQ.svgz" alt="\Lambda \subseteq \Z^n" title="\Lambda \subseteq \Z^n"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="16" src="https://math.fontein.de/formulae/Q8X_7IMZ1ugMeaxcLpoOFcNVOGvObyos0zkHdw.svgz" alt="floor" title="floor"></span> induces a map <span class="inline-formula"><img class="img-inline-formula img-formula" width="114" height="18" src="https://math.fontein.de/formulae/ezDX1zs_B0Oxw1nxBFaxeD4yWQHXmYEtZpA1BA.svgz" alt="\R^n/\Lambda \to \Z^n/\Lambda" title="\R^n/\Lambda \to \Z^n/\Lambda"></span>. Now, that <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> does not depends on fractions simply means that <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> factors through <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="16" src="https://math.fontein.de/formulae/Q8X_7IMZ1ugMeaxcLpoOFcNVOGvObyos0zkHdw.svgz" alt="floor" title="floor"></span>, i.e. that we can write <span class="inline-formula"><img class="img-inline-formula img-formula" width="139" height="17" src="https://math.fontein.de/formulae/CeSRWK1TFxKHHlvAweaVdHlbiJTGcoz_Djhzew.svgz" alt="red = red' \circ floor" title="red = red' \circ floor"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="132" height="18" src="https://math.fontein.de/formulae/PBjkn_5kwQiJO8emTZMeiVo0MwIjRFS52ww5jg.svgz" alt="red' : \Z^n/\Lambda \to X" title="red' : \Z^n/\Lambda \to X"></span>.
</p>
<p>
Moreover, if in the following we specify discrete infrastructures, we often just define <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> for values in <span class="inline-formula"><img class="img-inline-formula img-formula" width="43" height="18" src="https://math.fontein.de/formulae/IoA0BuM80DWvbyrrE5zoooQiZuyPkzKWW0aUjw.svgz" alt="\Z^n/\Lambda" title="\Z^n/\Lambda"></span>. In that case, for elements <span class="inline-formula"><img class="img-inline-formula img-formula" width="134" height="18" src="https://math.fontein.de/formulae/SqfdPeerp6a_.odVZ628BM6KQn5o2H143I7cjg.svgz" alt="v \in \R^n/\Lambda \setminus \Z^n/\Lambda" title="v \in \R^n/\Lambda \setminus \Z^n/\Lambda"></span>, define <span class="inline-formula"><img class="img-inline-formula img-formula" width="182" height="18" src="https://math.fontein.de/formulae/xNh3ezpEzYlYodQEtb_z0.wTSgDEPHvueO.HKw.svgz" alt="red(v) := red(floor(v))" title="red(v) := red(floor(v))"></span>.
</p>
<p>
In case <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="18" src="https://math.fontein.de/formulae/Q3Wi7WBtdKSnv1gaLvYnlftQgM.JrmfkFL4zUA.svgz" alt="(X, d, red)" title="(X, d, red)"></span> is discrete, consider the subset
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="464" height="20" src="https://math.fontein.de/formulae/YM76NrqV.ThBKY_h_GfwB6buGvQ5.8umAg9MKQ.svgz" alt="\fRep_{disc} := \fRep_{disc}(X, d, red) := \{ (x, f) \in \fRep \mid f \in \Z^n \}." title="\fRep_{disc} := \fRep_{disc}(X, d, red) := \{ (x, f) \in \fRep \mid f \in \Z^n \}.">
</div>
<p>
Then
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="224" height="23" src="https://math.fontein.de/formulae/tcrxpNF0qHVEFMEuPhG3SqFRVcB_Delchx2yGw.svgz" alt="\Psi|_{\fRep_{disc}} : \fRep_{disc} \to \Z^n/\Lambda" title="\Psi|_{\fRep_{disc}} : \fRep_{disc} \to \Z^n/\Lambda">
</div>
<p>
is an isomorphism.
</p>
</div>
</div>
<div class="subsection-block">
<h4>
Finite Abelian Groups as Infrastructures.
</h4>
<div>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span> be a finite abelian group, generated by <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="11" src="https://math.fontein.de/formulae/Fu0_nlKIc0UUBt4FnNjm2anOMMT5GEx2bVAbSA.svgz" alt="g_1, \dots, g_n" title="g_1, \dots, g_n"></span>. Consider the <em>relation lattice</em> <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="15" src="https://math.fontein.de/formulae/J8WZLQNTj4hsLiLptic9ZbHN6du478Qep2spzQ.svgz" alt="\Lambda \subseteq \Z^n" title="\Lambda \subseteq \Z^n"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="11" src="https://math.fontein.de/formulae/Fu0_nlKIc0UUBt4FnNjm2anOMMT5GEx2bVAbSA.svgz" alt="g_1, \dots, g_n" title="g_1, \dots, g_n"></span>, defined by
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="235" height="52" src="https://math.fontein.de/formulae/UYMavUU63zLuPsqTjsoh87WUDSEexQRlKibJRQ.svgz" alt="(v_1, \dots, v_n) \in \Lambda \Leftrightarrow \prod_{i=1}^n g_i^{v_i} = 1." title="(v_1, \dots, v_n) \in \Lambda \Leftrightarrow \prod_{i=1}^n g_i^{v_i} = 1.">
</div>
<p>
Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/XY_2LK590TGO0.jb0ZbkmaiBi4kJr1xvp52q2g.svgz" alt="\Lambda" title="\Lambda"></span> is the kernel of <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="12" src="https://math.fontein.de/formulae/OvmVVs0xwElKEGt057JVsho6qkpQM5sKP5N14g.svgz" alt="\Z^n \to G" title="\Z^n \to G"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="182" height="20" src="https://math.fontein.de/formulae/BICwYv67dMLULMbbQUwKQKl28GMRLtlEdGZQkw.svgz" alt="(v_1, \dots, v_n) \mapsto \prod_{i=1}^n g_i^{v_i}" title="(v_1, \dots, v_n) \mapsto \prod_{i=1}^n g_i^{v_i}"></span>, and
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="334" height="52" src="https://math.fontein.de/formulae/vNYGsJMYcKofWj_bu5_0DxQPGPX_d7x9oZI2Qw.svgz" alt="\varphi : \Z^n/\Lambda \to G, \quad (v_1, \dots, v_n) + \Lambda \mapsto \prod_{i=1}^n g_i^{v_i}" title="\varphi : \Z^n/\Lambda \to G, \quad (v_1, \dots, v_n) + \Lambda \mapsto \prod_{i=1}^n g_i^{v_i}">
</div>
<p>
is a group isomorphism. Define <span class="inline-formula"><img class="img-inline-formula img-formula" width="59" height="12" src="https://math.fontein.de/formulae/ub9MldwOl6ws9FT1tGUr_i5MqFAjBYd_m7_07g.svgz" alt="X := G" title="X := G"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="68" height="18" src="https://math.fontein.de/formulae/JhMNWD62vva7slnEGXf4.OAgqL8Iy6LwHgfcZQ.svgz" alt="d := \varphi^{-1}" title="d := \varphi^{-1}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="66" height="16" src="https://math.fontein.de/formulae/jxsJUomgXcUIpnv7cESAhTunDKoE48PiyW5m1w.svgz" alt="red := \varphi" title="red := \varphi"></span> (or, more precisely, <span class="inline-formula"><img class="img-inline-formula img-formula" width="125" height="16" src="https://math.fontein.de/formulae/vt9NvN1f2qmwPBK1Bl3MXr91ivZV8FT6ypsx8w.svgz" alt="red := \varphi \circ floor" title="red := \varphi \circ floor"></span>); then <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="18" src="https://math.fontein.de/formulae/Q3Wi7WBtdKSnv1gaLvYnlftQgM.JrmfkFL4zUA.svgz" alt="(X, d, red)" title="(X, d, red)"></span> is an <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructure. Moreover, for <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="17" src="https://math.fontein.de/formulae/viaMtbvDfwHuEUdUVjYivnoijvqaRVIdjhQBPA.svgz" alt="g, g' \in G" title="g, g' \in G"></span>, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="107" height="18" src="https://math.fontein.de/formulae/zDjZNIktpEwn5gRaOzktD8bNttmPlir0QI4JGQ.svgz" alt="\gs(g, g') = g g'" title="\gs(g, g') = g g'"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="11" src="https://math.fontein.de/formulae/oALhjAUt95L9B1GjlImGOqnVYMSkaj0JkVGFrg.svgz" alt="\gs" title="\gs"></span> equals the group operation of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span>. Hence, every finite abelian group can be seen in a natural way as an infrastructure.
</p>
<p>
Moreover, this shows that <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="12" src="https://math.fontein.de/formulae/DxlmDzniFM_09XzQnKCoRu8h2JBxVd5JriHzmg.svgz" alt="d" title="d"></span> can be thought of as an analogue to the discrete logarithm map, and <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span> is an analogue of the power map <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="15" src="https://math.fontein.de/formulae/qej5hFjgd7bFGsmrDphlLl5M_iRT2UuwuRhZmA.svgz" alt="n \mapsto g^n" title="n \mapsto g^n"></span>. In particular, we obtained the goal described in <a href="https://math.fontein.de/2009/07/20/the-discrete-logarithm-problem-and-generalizations/">the first post of this series</a>: we found a generalization of the discrete logarithm problem to a non-associative algebraic structure. In the next post, I will how such infrastructures can be obtained from global fields; this gives a rich source of examples for <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructures.
</p>
</div>
</div>
<div class="subsection-block">
<h4>
What about baby steps?
</h4>
<div>
<p>
Note that in the above discussion, I simply ignored baby steps. In the one-dimensional case, <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/eCr2xk03b8XPN3FP16iK5HS4b8Ql_90EmVknKw.svgz" alt="\R" title="\R"></span> has a canonical direction (namely the positive one) and so has <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="18" src="https://math.fontein.de/formulae/Blo1FniNoktf2EKQy0CWZxv_r_wmqWWcM_Pc9A.svgz" alt="\R/R\Z" title="\R/R\Z"></span>, whence saying “go to the next element” makes sense. Opposed to that, in <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/bZVXuWhECmRV9pmTW_AgHZeLTOYax7svHPDCxQ.svgz" alt="\R^n" title="\R^n"></span>, there are infinitely many directions, no one better than another. Even if we fix a direction, “go to the next element in that direction” seems to not really make sense. So far, I have not seen any definition of baby steps in this case which works for <em>all</em> <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="8" src="https://math.fontein.de/formulae/CiIJDoNXXhwwshmAknaOy.cbqWs.Z_qmDZe21A.svgz" alt="n" title="n"></span>-dimensional infrastructures.
</p>
<p>
Note that in the case of infrastructures obtained from global fields, one has some kind of canonical baby steps (even though there are still some choices left). In fact, there are <span class="inline-formula"><img class="img-inline-formula img-formula" width="41" height="13" src="https://math.fontein.de/formulae/fgRY_0KZ1v9Dol4ARGUMspWZdQu629FDAEQ9MA.svgz" alt="n + 1" title="n + 1"></span> of them. To define them, though, one needs more information than just <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="12" src="https://math.fontein.de/formulae/DxlmDzniFM_09XzQnKCoRu8h2JBxVd5JriHzmg.svgz" alt="d" title="d"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/x9uDaMcRWmFnhjgyf0CLBlJpQRUKxXbuUM0.1Q.svgz" alt="red" title="red"></span>: one needs information about a <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="18" src="https://math.fontein.de/formulae/AjCyaw2HagCBfGhDL8_wwlnYMLlETAC11TNcKw.svgz" alt="(n + 1)" title="(n + 1)"></span>-st dimension, both for constructing the reduction map and for defining baby steps.
</p>
</div>
</div>
</div>