Equidistribution of values of linear forms on quadratic surfaces

TL;DR

This paper demonstrates that, under specific algebraic conditions, the values of a linear map at integer points on a quadratic surface are evenly distributed, extending previous results with new quantitative methods based on unipotent flow equidistribution.

Contribution

It provides a new quantitative proof of equidistribution of linear forms on quadratic surfaces using unipotent flow techniques, under certain algebraic conditions.

Findings

Values are equidistributed under algebraic conditions

Quantitative bounds on distribution are established

Method extends previous theoretical results

Abstract

In this paper we investigate the distribution of the set of values of a linear map at integer points on a quadratic surface. In particular, it is shown that subject to certain algebraic conditions, this set is equidistributed. This can be thought of as a quantitative version of the main result from [2011arXiv1111.4428S]. The methods used are based on those developed by A. Eskin, S. Mozes and G. Margulis in [MR1609447]. Specifically, they rely on equidistribution properties of unipotent flows.