Poissonian correlations of $αn^d$ mod $1$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910217987424256 |
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| author | Lutsko, Chris Rome, Nick Technau, Niclas |
| author_facet | Lutsko, Chris Rome, Nick Technau, Niclas |
| contents | Let $x(n):=αn^d \mod 1$ for integer $d >1$ and non-zero real $α$. We show that $\{x(n)\}_{n>0}$ has Poissonian $\ell$-point correlations for almost all choices of $α$ when $d$ is large (depending on $\ell$). This falls in line with the expected behavior from the Berry--Tabor conjecture. Further, in the spirit of a conjecture of Rudnick--Sarnak, we show Poissonian $\ell$-point correlations for a set of badly approximable $α$ of full Hausdorff dimension by a Fourier analytic transference principle.
The proof makes use of an application of the determinant method to count points on a diagonal hypersurface of degree $d$ in such a way as to capture the contribution of points belonging to lower dimensional varieties. As $d$ grows, these `special solutions' dominate the count and non-special solutions become increasingly rare. This stratified counting statement allows us to control the number of points on average very effectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06974 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Poissonian correlations of $αn^d$ mod $1$ Lutsko, Chris Rome, Nick Technau, Niclas Number Theory 11K06, 11L07, 15B48 Let $x(n):=αn^d \mod 1$ for integer $d >1$ and non-zero real $α$. We show that $\{x(n)\}_{n>0}$ has Poissonian $\ell$-point correlations for almost all choices of $α$ when $d$ is large (depending on $\ell$). This falls in line with the expected behavior from the Berry--Tabor conjecture. Further, in the spirit of a conjecture of Rudnick--Sarnak, we show Poissonian $\ell$-point correlations for a set of badly approximable $α$ of full Hausdorff dimension by a Fourier analytic transference principle. The proof makes use of an application of the determinant method to count points on a diagonal hypersurface of degree $d$ in such a way as to capture the contribution of points belonging to lower dimensional varieties. As $d$ grows, these `special solutions' dominate the count and non-special solutions become increasingly rare. This stratified counting statement allows us to control the number of points on average very effectively. |
| title | Poissonian correlations of $αn^d$ mod $1$ |
| topic | Number Theory 11K06, 11L07, 15B48 |
| url | https://arxiv.org/abs/2605.06974 |