Equations and character sums with matrix powers, Kloosterman sums over small subgroups and quantum ergodicity

Abstract

We obtain a nontrivial bound on the number of solutions to the equation $$A^{x_1}+\dots +A^{x_{\nu }}=A^{x_{\nu +1}}+\dots +A^{x_{2\nu }},1\le x_1,\dots ,x_{2\nu }\le \tau ,$$ with a fixed $n\times n$ matrix $A$ over a finite field ${\mathbb{F}}_q$ of $q$ elements of multiplicative order $\tau$. We give applications of our result to obtaining a new bound of additive character sums with a matrix exponential function, which is nontrivial beyond the square-root threshold. For $n=2$ this equation has been considered by Kurlberg and Rudnick (2001) (for $\nu =2$) and Bourgain (2005) (for large $\nu$) in their study of quantum ergodicity for linear maps over residue rings. Here we use a new approach to improve their results. We also obtain a bound on Kloosterman sums over small subgroups, of size below the square-root threshold.

This paper supersedes our previous submission: arXiv:2108.13146 which is now obsolete