On discrepancy, intrinsic Diophantine approximation, and spectral gaps

TL;DR

This paper links spectral gap bounds for algebraic group actions on homogeneous spaces with discrepancy estimates for rational points, providing effective bounds for property τ in certain arithmetic lattices.

Contribution

It develops new techniques for bounding operator norms related to spectral gaps and connects these bounds to intrinsic Diophantine approximation on algebraic varieties.

Findings

Established bounds for spectral gaps of algebraic group actions.

Linked spectral gap bounds to discrepancy estimates for rational points.

Provided effective bounds for property τ in specific arithmetic lattices.

Abstract

In the present paper we establish bounds for the size of the spectral gap for actions of algebraic groups on certain homogeneous spaces. Our approach is based on estimating operator norms of suitable averaging operators, and we develop techniques for establishing both upper and lower bounds for such norms. We shall show that this analytic problem is closely related to the arithmetic problem of establishing bounds on the discrepancy of distribution for rational points on algebraic group varieties. As an application, we show how to establish an effective bound for property $\tau$ of congruence subgroups of arithmetic lattices in algebraic groups which are forms of $SL(2)$, using estimates in intrinsic Diophantine approximation which follow from Heath-Brown's analysis of rational points on 3-dimensional quadratic surfaces.