On quantum ergodicity for higher‐dimensional cat maps modulo prime powers

Abstract A discrete model of quantum ergodicity of linear maps generated by symplectic matrices modulo an integer , has been studied for and almost all by Kurlberg and Rudnick (2001, Comm. Math. Phys., 222, 201–227). Their result has been strengthened by Bourgain and then by Ostafe, Shparlinski and Voloch (2023, Int. Math. Res. Not., 2023, 14196–14238). For arbitrary , this has been studied by Kurlberg, Ostafe, Rudnick and Shparlinski. The corresponding equidistribution results, for certain eigenfunctions, share the same feature: they apply to almost all moduli and are unable to provide an explicit construction of such ‘good’ values of . Here, using a bound of Shparlinski (1978, Proc. Voronezh State Pedagogical Inst., 197, 74–85) on exponential sums with linear recurrence sequences modulo a power of a fixed prime, we construct such an explicit sequence of , with a power saving on the discrepancy.