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

TL;DR

This paper establishes new bounds on solutions to matrix power equations over finite fields, leading to improved estimates for character sums and Kloosterman sums, with applications to quantum ergodicity and exponential sums.

Contribution

It introduces a novel approach to bounding solutions of matrix power equations, extending previous results and improving bounds for character and Kloosterman sums over small subgroups.

Findings

Derived nontrivial bounds on solutions to matrix power equations over finite fields.

Improved bounds on additive character sums with matrix exponential functions.

Established bounds on Kloosterman sums over subgroups smaller than the square-root threshold.

Abstract

We obtain a nontrivial bound on the number of solutions to the equation $$ A^{x_1} + \ldots + A^{x_\nu} = A^{x_{\nu+1}} + \ldots + A^{x_{2\nu}}, \quad 1 \le x_1, \ldots,x_{2\nu} \le \tau, $$ 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.