A uniform Linnik basic lemma and entropy bounds

TL;DR

This paper proves a uniform version of Linnik's basic lemma using theta-series and geometric invariant theory, and applies it to establish entropy bounds for invariant measures on certain homogeneous toral sets in GL(4).

Contribution

It introduces a uniform version of Linnik's basic lemma and applies it to derive entropy bounds for invariant measures in specific algebraic settings.

Findings

Established entropy bounds for limits of invariant measures.

Proved a uniform version of Linnik's basic lemma.

Applied geometric invariant theory in the context of theta-series.

Abstract

We prove a version of Linnik's basic lemma uniformly over the base field using theta-series and geometric invariant theory in the spirit of Khayutin's approach (Duke Math. J., 168(12), 2019). As an application, we establish entropy bounds for limits of invariant measures on homogeneous toral sets in GL(4) of biquadratic, cyclic, or dihedral type.