标题： 关于模素数幂的高维猫映射的量子遍历性 显示英文标题

标题： On quantum ergodicity for higher dimensional cat maps modulo prime powers

摘要： 一个由辛矩阵生成的线性映射的量子遍历性的离散模型，对模整数$A \in \mathrm{Sp}(2d,\mathbb{Z})$的$N\ge 1$，由 P. Kurlberg 和 Z. Rudnick (2001) 研究了$d=1$和几乎所有$N$。他们的结果被 J. Bourgain (2005) 加强，并随后由 A. Ostafe, I. E. Shparlinski 和 J. F. Voloch (2023) 进一步加强。对于任意的$d$，这已被 P. Kurlberg, A. Ostafe, Z. Rudnick 和 I. E. Shparlinski (2024) 研究。对于某些特征函数的相应均匀分布结果具有相同的特性：它们适用于几乎所有模数$N$，并且无法提供这样的“良好”值$N$的显式构造。 在这里，利用I. E. Shparlinski（1978）关于模固定素数幂的线性递推序列的指数和的界，我们构造了这样一个显式序列$N$，其偏差有幂次节省。 显示英文摘要

摘要： A discrete model of quantum ergodicity of linear maps generated by symplectic matrices $A \in \mathrm{Sp}(2d,\mathbb{Z})$ modulo an integer $N\ge 1$, has been studied for $d=1$ and almost all $N$ by P. Kurlberg and Z. Rudnick (2001). Their result has been strengthened by J. Bourgain (2005) and subsequently by A. Ostafe, I. E. Shparlinski, and J. F. Voloch (2023). For arbitrary $d$ this has been studied by P. Kurlberg, A. Ostafe, Z. Rudnick and I. E. Shparlinski (2024). The corresponding equidistribution results, for certain eigenfunctions, share the same feature: they apply to almost all moduli $N$ and are unable to provide an explicit construction of such ``good'' values of $N$. Here, using a bound of I. E. Shparlinski (1978) on exponential sums with linear recurrence sequences modulo a power of a fixed prime, we construct such an explicit sequence of $N$, with a power saving on the discrepancy.