Integral points of bounded degree on the projective line and in dynamical orbits
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- by Joseph Gunther and Wade Hindes PDF
- Proc. Amer. Math. Soc. 145 (2017), 5087-5096 Request permission
Abstract:
Let $D$ be a non-empty effective divisor on $\mathbb {P}^1$. We show that when ordered by height, any set of $(D,S)$-integral points on $\mathbb {P}^1$ of bounded degree has relative density zero. We then apply this to arithmetic dynamics: let $\varphi (z)\in \overline {\mathbb {Q}}(z)$ be a rational function of degree at least two whose second iterate $\varphi ^2(z)$ is not a polynomial. We show that as we vary over points $P\in \mathbb {P}^1(\overline {\mathbb {Q}})$ of bounded degree, the number of algebraic integers in the forward orbit of $P$ is absolutely bounded and zero on average.References
- Fabrizio Barroero, Algebraic $S$-integers of fixed degree and bounded height, Acta Arith. 167 (2015), no. 1, 67–90. MR 3310491, DOI 10.4064/aa167-1-4
- Enrico Bombieri and Walter Gubler, Heights in Diophantine geometry, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR 2216774, DOI 10.1017/CBO9780511542879
- Antoine Chambert-Loir and Yuri Tschinkel, Integral points of bounded height on partial equivariant compactifications of vector groups, Duke Math. J. 161 (2012), no. 15, 2799–2836. MR 2999313, DOI 10.1215/00127094-1813638
- S. D. Cohen, The distribution of Galois groups and Hilbert’s irreducibility theorem, Proc. London Math. Soc. (3) 43 (1981), no. 2, 227–250. MR 628276, DOI 10.1112/plms/s3-43.2.227
- P. Corvaja and U. Zannier, On integral points on surfaces, Ann. of Math. (2) 160 (2004), no. 2, 705–726. MR 2123936, DOI 10.4007/annals.2004.160.705
- David S. Dummit and Richard M. Foote, Abstract algebra, 3rd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2004. MR 2286236
- Leonhard Euler, Variae observationes circa series infinitas, Commentarii academiae scientiarum Petropolitanae, 9:160–188, 1744.
- Wade Hindes, The average number of integral points in orbits, September 2015. Preprint. arXiv:1509.00752.
- Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
- Aaron Levin, Generalizations of Siegel’s and Picard’s theorems, Ann. of Math. (2) 170 (2009), no. 2, 609–655. MR 2552103, DOI 10.4007/annals.2009.170.609
- Aaron Levin, Integral points of bounded degree on affine curves, Compos. Math. 152 (2016), no. 4, 754–768. MR 3484113, DOI 10.1112/S0010437X15007708
- David Masser and Jeffrey D. Vaaler, Counting algebraic numbers with large height. II, Trans. Amer. Math. Soc. 359 (2007), no. 1, 427–445. MR 2247898, DOI 10.1090/S0002-9947-06-04115-8
- David Masser and Jeffrey D. Vaaler, Counting algebraic numbers with large height. I, Diophantine approximation, Dev. Math., vol. 16, SpringerWienNewYork, Vienna, 2008, pp. 237–243. MR 2487698, DOI 10.1007/978-3-211-74280-8_{1}4
- Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, 3rd ed., Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1997. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt; With a foreword by Brown and Serre. MR 1757192, DOI 10.1007/978-3-663-10632-6
- Joseph H. Silverman, Integer points, Diophantine approximation, and iteration of rational maps, Duke Math. J. 71 (1993), no. 3, 793–829. MR 1240603, DOI 10.1215/S0012-7094-93-07129-3
- Joseph H. Silverman, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol. 241, Springer, New York, 2007. MR 2316407, DOI 10.1007/978-0-387-69904-2
- Xiangjun Song and Thomas J. Tucker, Dirichlet’s theorem, Vojta’s inequality, and Vojta’s conjecture, Compositio Math. 116 (1999), no. 2, 219–238. MR 1686848, DOI 10.1023/A:1000948001301
- Paul Vojta, A generalization of theorems of Faltings and Thue-Siegel-Roth-Wirsing, J. Amer. Math. Soc., 5(4):763–804, 1992.
- Paul Vojta, Diophantine approximation and Nevanlinna theory, Arithmetic geometry, Lecture Notes in Math., vol. 2009, Springer, Berlin, 2011, pp. 111–224. MR 2757629, DOI 10.1007/978-3-642-15945-9_{3}
- Eduard A. Wirsing, On approximations of algebraic numbers by algebraic numbers of bounded degree. In 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pages 213–247. Amer. Math. Soc., Providence, R.I., 1971.
Additional Information
- Joseph Gunther
- Affiliation: Department of Mathematics, The Graduate Center, City University of New York (CUNY), 365 Fifth Avenue, New York, New York 10016
- MR Author ID: 1171459
- Email: JGunther@gradcenter.cuny.edu
- Wade Hindes
- Affiliation: Department of Mathematics, The Graduate Center, City University of New York (CUNY), 365 Fifth Avenue, New York, New York 10016
- MR Author ID: 1022776
- Email: whindes@gc.cuny.edu
- Received by editor(s): August 4, 2016
- Received by editor(s) in revised form: December 5, 2016, and January 3, 2017
- Published electronically: June 8, 2017
- Additional Notes: The first author was partially supported by National Science Foundation grant DMS-1301690
- Communicated by: Mathew A. Papanikolas
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 5087-5096
- MSC (2010): Primary 11D45, 37P15; Secondary 11G50, 11R04, 14G05
- DOI: https://doi.org/10.1090/proc/13653
- MathSciNet review: 3717939