On quantum ergodicity for higher dimensional cat maps modulo prime powers

TL;DR

This paper advances the understanding of quantum ergodicity for higher-dimensional linear maps by explicitly constructing sequences of moduli with improved discrepancy bounds, extending prior results that only applied to almost all moduli.

Contribution

It provides an explicit construction of moduli sequences with better discrepancy bounds for quantum ergodicity in higher dimensions, building on previous almost-all results.

Findings

Constructed explicit sequences of moduli with power savings on discrepancy

Extended quantum ergodicity results to higher dimensions for specific moduli

Improved bounds on exponential sums with linear recurrence sequences

Abstract

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.