On quantum ergodicity for higher dimensional cat maps modulo prime powers
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911152322117632 |
|---|---|
| author | Bhakta, Subham Shparlinski, Igor E. |
| author_facet | Bhakta, Subham Shparlinski, Igor E. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00325 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On quantum ergodicity for higher dimensional cat maps modulo prime powers Bhakta, Subham Shparlinski, Igor E. Number Theory Mathematical Physics Dynamical Systems 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. |
| title | On quantum ergodicity for higher dimensional cat maps modulo prime powers |
| topic | Number Theory Mathematical Physics Dynamical Systems |
| url | https://arxiv.org/abs/2507.00325 |