A uniform Linnik basic lemma and entropy bounds

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 (2019)]. As an application, we establish entropy bounds for weak*-limits of invariant measures on homogeneous toral sets in ${\mathbf{GL}}_4$ of biquadratic, cyclic, or dihedral type.