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Distribution of integral division points on the algebraic torus. (English) Zbl 1416.11113
Let \(K\) be a number field, \(S\) a finite set of places of \(K\) including the Archimedean ones, \(\overline{K}\) an algebraic closure of \(K\), \(\Gamma\) a subgroup of \({\mathbb G}_m(\overline{K})\) of finite rank, \(D\) a nonzero effective divisor on \({\mathbb G}_m\).
The main result of the paper under review is the following. Let \((I_n)_{n\geq 1}\) be a decreasing sequence of closed intervals of \([0,\infty)\) with intersection \(I\). If the set of \(\gamma\in \Gamma\) which are \(S\)-integral in \({\mathbb G}_m\) with logarithmic Weil height \({\mathrm h}(\gamma)\) in \(I_n\) is infinite for all \(n\geq 1\), then \(D\) is the translate of a torsion divisor on \({\mathbb G}_m\) by an element \(\alpha\in\Gamma\) which is \(S\)-integral on \({\mathbb G}_m\) relative to \(D\) and with \({\mathrm h}(\gamma)\in I\).
This result is equivalent to the next one: if \((\gamma_n)_{n\geq 1}\) is a sequence of pairwise distinct elements of \(\Gamma\) that are \(S\)-integral on \({\mathbb G}_m\) relative to \(D\), then \(D\) is the translate of a torsion divisor on \({\mathbb G}_m\) by some \(\alpha\in\Gamma\) that is \(S\)-integral on \({\mathbb G}_m\) relative to \(D\) and with \(\lim_{n\to\infty}{\mathrm h}(\gamma_n\alpha^{-1})=0\). This generalizes [M. Baker et al., Algebra Number Theory 2, No. 2, 217–248 (2008; Zbl 1182.11030)] which deals with the special case where \(\Gamma\) is the group of roots of unity. It also generalizes [D. Grant and S.-I. Ih, Compos. Math. 149, No. 12, 2011–2035 (2013; Zbl 1292.11072)].
The proof uses lower bounds for linear forms in logarithms of algebraic numbers, both in the Archimedean case (A. Baker) and in the ultrametric one (Yu Kunrui), an explicit version of Weyl’s equidistribution criterion due to Erdős and Turán and an auxiliary result due to H. P. Schlickewei and W. M. Schmidt [Acta Arith. 72, No. 1, 1–44 (1995; Zbl 0851.11007)] based on the geometry of numbers.
The authors also extend their result from \({\mathbb G}_m\) to the projective line \({\mathbb P}_1\). They propose some extension to semi abelian varieties and also an analogous conjecture for dynamical systems.
MSC:
11G50 Heights
11J61 Approximation in non-Archimedean valuations
11J71 Distribution modulo one
11J86 Linear forms in logarithms; Baker’s method
11L15 Weyl sums
14G25 Global ground fields in algebraic geometry
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
20G30 Linear algebraic groups over global fields and their integers
37P05 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps
37P35 Arithmetic properties of periodic points
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