On quantum ergodicity for higher dimensional cat maps modulo prime powers

A discrete model of quantum ergodicity of linear maps generated by symplectic matrices A ∈ Sp(2d,Z) modulo an integer N≥ 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.