A note on logarithmic equidistribution

TL;DR

This paper investigates the distribution of algebraic numbers near points on the unit circle, establishing new results on their heights and averages, and extends these findings to the p-adic context.

Contribution

It proves the existence of algebraic number sequences with specific height and average properties for functions related to points on the unit circle, completing prior characterizations.

Findings

Existence of algebraic number sequences with bounded averages

Extension of results to p-adic setting

Characterization of functions $f____________________________________________________

Abstract

For every algebraic number $\kappa$ on the unit circle which is not a root of unity we prove the existence of a strict sequence of algebraic numbers whose height tends to zero, such that the averages of the evaluation of $f_\kappa(z)=\log|z -\kappa|$ in the conjugates are essentially bounded from above by $-h(\kappa)$. This completes a characterisation on functions $f_\kappa$ initiated by Autissier and Baker-Masser, who cover the cases $\kappa=2$ and $|\kappa|\ne 1$ respectively. Using the same ideas we also prove analogues in the $p$-adic setting.