Equations and Character Sums with Matrix Powers

Wednesday, 15 March 2023

Abstract

We obtain a nontrivial bound on the number of solutions to the equation 

$$\sum_{i=1}^{\nu} A^{x_i} = \sum_{i=\nu+1}^{2\nu} A^{x_i}, \qquad 1 \le x_i \le \tau,$$ with a fixed $n\times n$ matrix $A$ over a finite field $\mathbb{F}_q$ of $q$ elements of multiplicative order $\tau$. We apply our result to obtain a new bound for additive character sums with a matrix exponential function and Kloosterman sums over small subgroups, nontrivial beyond the square-root threshold. 

For $n=2$ this equation has been considered by Kurlberg and Rudnick (for $\nu=2$) and Bourgain (for large $\nu$) in their study of quantum ergodicity for a linear map over residue rings.  Using our approach, very different from theirs, we obtain a bound with an explicit saving in the uniformity of distribution result for eigenvalues of this map

(joint work with A. Ostafe and J. F. Voloch)