On totally real numbers and equidistribution

TL;DR

This paper extends previous work on the Weil height spectrum of totally real numbers by establishing a lower bound for all such numbers, not just integers, using equidistribution techniques.

Contribution

It introduces a new lower bound for the height of all totally real numbers, removing the integer restriction from prior studies.

Findings

Established a lower bound on the Weil height for all totally real numbers.

Applied a quantitative equidistribution theorem to prove the bound.

Extended previous results from integers to all totally real numbers.

Abstract

C.J. Smyth and later Flammang studied 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. We remove the hypothesis that we consider only integers and establish an lower bound on the limit infimum of the height for all totally real numbers. Our proof relies on a quantitative equidistribution theorem for numbers of small height.