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

TL;DR

This paper extends the understanding of the distribution and minimal heights of algebraic numbers in totally real and p-adic fields, using equidistribution results to generalize Bogomolov-type theorems in arithmetic dynamics.

Contribution

It generalizes Bogomolov-type results for various heights in arithmetic dynamics, building on classical work on totally real and p-adic numbers.

Findings

Established bounds for heights in totally real and p-adic fields.

Generalized Bogomolov property to new classes of heights.

Connected equidistribution of low-height points to these bounds.

Abstract

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.