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

Volume 170 / 2015

Paul Fili, Zachary Miner Acta Arithmetica 170 (2015), 15-25 MSC: 11G50, 11R80, 37P30. DOI: 10.4064/aa170-1-2


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.


  • Paul FiliDepartment of Mathematics
    Oklahoma State University
    Stillwater, OK 74078, U.S.A.
  • Zachary MinerDepartment of Mathematics
    University of Texas at Austin
    Austin, TX 78712, U.S.A.

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