Quantitative equidistribution of Galois orbits of small points in the N-dimensional torus

TL;DR

This paper provides a quantitative version of Bilu's theorem, offering explicit bounds on how Galois orbits of small points in the N-dimensional torus distribute compared to uniform distribution, based on height and degree.

Contribution

It introduces a quantitative bound for the discrepancy between Galois orbits and uniform distribution in the N-dimensional torus, extending Bilu's theorem.

Findings

Explicit discrepancy bounds in terms of height and degree

Quantitative analysis of Galois orbit distribution

Extension of Bilu's theorem to higher dimensions

Abstract

We present a quantitative version of Bilu's theorem on the limit distribution of Galois orbits of sequences of points of small height in the $N$-dimensional algebraic torus. Our result gives, for a given point, an explicit bound for the discrepancy between its Galois orbit and the uniform distribution on the compact subtorus, in terms of the height and the generalized degree of the point.