Additive energy of cyclic matrix groups and character sums with matrix exponential functions

TL;DR

This paper establishes a new bound on solutions to a matrix exponential equation over finite fields, with applications to character sums and additive problems involving matrices, especially for special matrix groups.

Contribution

It provides the first nontrivial bounds for solutions to matrix exponential equations over finite fields and applies these results to character sums and additive matrix problems.

Findings

Bound on solutions to matrix exponential equations over finite fields

Application to nontrivial bounds on additive character sums

Results are particularly strong for SL(n,q) matrices with irreducible characteristic polynomial

Abstract

We obtain a nontrivial bound on the number of solutions to the equation $A^{x_1} + A^{x_2} = A^{x_3} + A^{x_4}$, $1 \le x_1,x_2,x_3,x_4 \le \tau$, with a fixed $n\times n$ matrix $A$ over a finite field ${\mathbb F}_q$ of $q$ elements of multiplicative order $\tau$. For $n=2$ this equation has been considered by Kurlberg and Rudnick (2001) in their study of quantum ergodicity for linear maps over ${\mathbb F}_q$. Furthermore, its multivariate analogue (also with $n=2$) has been studied by Bourgain (2005). 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, and also to a certain additive problem with matrices. Our results are especially strong for ${\rm SL}(n,q)$ matrices with an irreducible characteristic polynomial.