Effective equidistribution of Galois orbits for mildly regular test functions

TL;DR

This paper advances the understanding of how Galois orbits of small-height points in algebraic tori distribute effectively, emphasizing the role of test function regularity and extending Fourier analysis methods.

Contribution

It provides new effective bounds for Galois orbit equidistribution, generalizing previous results through a Fourier analysis framework that accounts for test function regularity.

Findings

Quantitative bounds depend on test function regularity

Extended Fourier analysis framework for equidistribution

Generalization of previous effective equidistribution results

Abstract

In this paper we provide a detailed study on effective versions of the celebrated Bilu's equidistribution theorem for Galois orbits of sequences of points of small height in the $N$-dimensional algebraic torus, identifying the quantitative dependence of the convergence in terms of the regularity of the test functions considered. We develop a general Fourier analysis framework that extends previous results obtained by Petsche (2005), and by D'Andrea, Narv\'aez-Clauss and Sombra (2017).