Effective equidistribution of rational points on expanding horospheres

TL;DR

This paper develops an effective method using harmonic analysis and Weil's bound to quantify the rate of equidistribution of rational points on expanding horospheres in three-dimensional Euclidean lattices.

Contribution

It introduces an alternative approach to measure classification, providing explicit convergence rates in dimension three.

Findings

Effective convergence rate estimate in dimension three

Utilizes harmonic analysis and Weil's bound for Kloosterman sums

Advances understanding of rational points distribution on horospheres

Abstract

Einsiedler, Mozes, Shah and Shapira [Compos. Math. 152 (2016), 667-692] prove an equidistribution theorem for rational points on expanding horospheres in the space of d-dimensional Euclidean lattices, with d>2. Their proof exploits measure classification results, but provides no insight into the rate of convergence. We pursue here an alternative approach, based on harmonic analysis and Weil's bound for Kloosterman sums, which in dimension d=3 yields an effective estimate on the rate of convergence.