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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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##### References:
 [1] Amoroso, Francesco; Zannier, Umberto, A relative Dobrowolski lower bound over abelian extensions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29, 3, 711-727 (2000) · Zbl 1016.11026 [2] Autissier, Pascal, Sur une question d’\'equir\'epartition de nombres alg\'ebriques, C. R. Math. Acad. Sci. Paris, 342, 9, 639-641 (2006) · Zbl 1129.11046 [3] Baker, A.; W\"ustholz, G., Logarithmic forms and group varieties, J. Reine Angew. Math., 442, 19-62 (1993) · Zbl 0788.11026 [4] Baker, Matthew; Ih, S.; Rumely, Robert, A finiteness property of torsion points, Algebra Number Theory, 2, 2, 217-248 (2008) · Zbl 1182.11030 [5] Bilu, Yuri, Limit distribution of small points on algebraic tori, Duke Math. J., 89, 3, 465-476 (1997) · Zbl 0918.11035 [6] Bombieri, Enrico; Gubler, Walter, Heights in Diophantine geometry, New Mathematical Monographs 4, xvi+652 pp. (2006), Cambridge University Press, Cambridge · Zbl 1115.11034 [7] Chambert-Loir, Antoine, Points de petite hauteur sur les vari\'et\'es semi-ab\'eliennes, Ann. Sci. \'Ecole Norm. Sup. (4), 33, 6, 789-821 (2000) · Zbl 1018.11034 [8] David, Sinnou; Philippon, Patrice, Sous-vari\'et\'es de torsion des vari\'et\'es semi-ab\'eliennes, C. R. Acad. Sci. Paris S\'er. I Math., 331, 8, 587-592 (2000) · Zbl 0972.11059 [9] Dobrowolski, E., On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith., 34, 4, 391-401 (1979) · Zbl 0416.12001 [10] Grant, David; Ih, S., Integral division points on curves, Compos. Math., 149, 12, 2011-2035 (2013) · Zbl 1292.11072 [11] Hardy, G. H.; Wright, E. M., An introduction to the theory of numbers, xxii+621 pp. (2008), Oxford University Press, Oxford · Zbl 1159.11001 [12] Harman, Glyn, Metric number theory, London Mathematical Society Monographs. New Series 18, xviii+297 pp. (1998), The Clarendon Press, Oxford University Press, New York · Zbl 1081.11057 [13] Hindry, Marc; Silverman, Joseph H., Diophantine geometry, Graduate Texts in Mathematics 201, xiv+558 pp. (2000), Springer-Verlag, New York · Zbl 0948.11023 [14] Ih, S., Integral points on the Chebyshev dynamical systems, J. Korean Math. Soc., 52, 5, 955-964 (2015) · Zbl 1392.37105 [15] Neukirch, J\"urgen, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 322, xviii+571 pp. (1999), Springer-Verlag, Berlin · Zbl 0956.11021 [16] Schlickewei, Hans Peter, Lower bounds for heights on finitely generated groups, Monatsh. Math., 123, 2, 171-178 (1997) · Zbl 0973.11067 [17] Schlickewei, H. P.; Schmidt, Wolfgang M., On polynomial-exponential equations, Math. Ann., 296, 2, 339-361 (1993) · Zbl 0805.11029 [18] Schmidt, Wolfgang M., Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics 1467, viii+217 pp. (1991), Springer-Verlag, Berlin · Zbl 0754.11020 [19] Silverman, Joseph H., Integer points, Diophantine approximation, and iteration of rational maps, Duke Math. J., 71, 3, 793-829 (1993) · Zbl 0811.11052 [20] Silverman, Joseph H., The arithmetic of dynamical systems, Graduate Texts in Mathematics 241, x+511 pp. (2007), Springer, New York · Zbl 1130.37001 [21] Waldschmidt, Michel, Diophantine approximation on linear algebraic groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 326, xxiv+633 pp. (2000), Springer-Verlag, Berlin · Zbl 0944.11024 [22] Yu, Kun Rui, Linear forms in $$p$$-adic logarithms. III, Compositio Math., 91, 3, 241-276 (1994) · Zbl 0819.11025 [23] Zorzitto, Frank, Discretely normed abelian groups, Aequationes Math., 29, 2-3, 172-174 (1985) · Zbl 0583.20039
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