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We obtain a nontrivial bound on the number of solutions to the equation   $$ \begin{align*} &amp;\sum_{i=1}^{\nu} A^{x_i} = \sum_{i=\nu+1}^{2\nu} A^{x_i}, \qquad 1 \leqslant x_i \leqslant \tau, \end{align*}$$  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, 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 linear maps over residue rings. We use a new approach to improve their results and also obtain a bound on Kloosterman sums over small subgroups, of size below the square-root threshold.

Equations and Character Sums with Matrix Powers, Kloosterman Sums over Small Subgroups, and Quantum Ergodicity