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

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.