Equidistribution and the heights of totally real and totally $p$-adic numbers

Paul Fili¹ ; Zachary Miner²

¹ Department of Mathematics Oklahoma State University Stillwater, OK 74078, U.S.A.
² Department of Mathematics University of Texas at Austin Austin, TX 78712, U.S.A.

Acta Arithmetica, Tome 170 (2015) no. 1, pp. 15-25

Résumé

C. J. Smyth was among the first to study the spectrum of the Weil height in the field of all totally real numbers, establishing both lower and upper bounds for the limit infimum of the height of all totally real integers, and determining isolated values of the height. Later, Bombieri and Zannier established similar results for totally $p$-adic numbers and, inspired by work of Ullmo and Zhang, termed this the Bogomolov property. In this paper, we use results on equidistribution of points of low height to generalize both Bogomolov-type results to a wide variety of heights arising in arithmetic dynamics.

Détail

DOI : 10.4064/aa170-1-2

Keywords: smyth among first study spectrum weil height field totally real numbers establishing lower upper bounds limit infimum height totally real integers determining isolated values height later bombieri zannier established similar results totally p adic numbers inspired work ullmo zhang termed bogomolov property paper results equidistribution points low height generalize bogomolov type results wide variety heights arising arithmetic dynamics