We obtain a nontrivial bound on the number of solutions to the equation $$ \begin{align*} &\sum_{i=1}^{\nu} A^{x_i} = \sum_{i=\nu+1}^{2\nu} A^{x_i}, \qquad 1 \leqslant x_i \leqslant \tau, \end{align*}$$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.

中文翻译：

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具有矩阵幂的方程和字符和、小子群上的 Kloosterman 和和量子遍历

我们得到方程解数的非平凡界限 $$ \begin{align*} &\sum_{i=1}^{\nu} A^{x_i} = \sum_{i=\nu+1} ^{2\nu} A^{x_i}, \qquad 1 \leqslant x_i \leqslant \tau, \end{align*}$$在有限域${ {\mathbb {F}}}_q$ 个乘法阶 $\tau $ 的 $q$ 个元素。我们应用我们的结果来获得具有矩阵指数函数的加性字符和的新界限，超出平方根阈值是非平凡的。对于 $n=2$，Kurlberg 和 Rudnick（对于 $\nu =2$）和 Bourgain（对于大 $\nu $）在他们研究残差环上线性映射的量子遍历性时考虑了这个方程。我们使用一种新方法来改进他们的结果，并在大小低于平方根阈值的小子组上获得 Kloosterman 和的界限。