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)

Speaker

Igor Shparlinski

Research area

Number Theory

Affilation

UNSW, Sydney

Date

Wednesday 15 March 2023, 2.00 pm

Location

RC-4082