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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.00708 |
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| _version_ | 1866912303618719744 |
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| author | Aistleitner, Christoph Yesha, Nadav |
| author_facet | Aistleitner, Christoph Yesha, Nadav |
| contents | Let $(x_n)_{n=1}^\infty$ be a sequence of integers. We study the number variance of dilations $(αx_n)_{n=1}^\infty$ modulo 1 in intervals of length $S$, and establish pseudorandom (Poissonian) behavior for Lebesgue-almost all $α$ throughout a large range of $S$, subject to certain regularity assumptions imposed upon $(x_n)_{n=1}^\infty$. For the important special case $x_n = p(n)$, where $p$ is a polynomial with integer coefficients of degree at least 2, we prove that the number variance is Poissonian for almost all $α$ throughout the range $0 \leq S \leq (\log N)^{-c}$, for a suitable absolute constant $c>0$. For more general sequences $(x_n)_{n=1}^\infty$, we give a criterion for Poissonian behavior for generic $α$ which is formulated in terms of the additive energy of the finite truncations $(x_n)_{n=1}^N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00708 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost sure asymptotics for the number variance of dilations of integer sequences Aistleitner, Christoph Yesha, Nadav Number Theory Probability Let $(x_n)_{n=1}^\infty$ be a sequence of integers. We study the number variance of dilations $(αx_n)_{n=1}^\infty$ modulo 1 in intervals of length $S$, and establish pseudorandom (Poissonian) behavior for Lebesgue-almost all $α$ throughout a large range of $S$, subject to certain regularity assumptions imposed upon $(x_n)_{n=1}^\infty$. For the important special case $x_n = p(n)$, where $p$ is a polynomial with integer coefficients of degree at least 2, we prove that the number variance is Poissonian for almost all $α$ throughout the range $0 \leq S \leq (\log N)^{-c}$, for a suitable absolute constant $c>0$. For more general sequences $(x_n)_{n=1}^\infty$, we give a criterion for Poissonian behavior for generic $α$ which is formulated in terms of the additive energy of the finite truncations $(x_n)_{n=1}^N$. |
| title | Almost sure asymptotics for the number variance of dilations of integer sequences |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2504.00708 |