What question did this study set out to answer?

The study aims to enhance understanding of quantum ergodicity in linear maps modulo prime powers and to construct explicit sequences with improved discrepancy.

February 20, 2026Open Access

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

Key Points

The study aims to enhance understanding of quantum ergodicity in linear maps modulo prime powers and to construct explicit sequences with improved discrepancy.
Analyzed discrete models of quantum ergodicity for certain linear maps.
Utilized results from prior research on equidistribution and eigenfunctions.
Applied a bound on exponential sums with linear recurrence sequences under prime powers to derive new sequences.
Developed an explicit sequence for moduli that showcases improved discrepancy features.
Confirmed that the equidistribution results apply to almost all moduli.
Highlighted limitations of existing approaches to find 'good' values of moduli.

Abstract

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.

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