Effective pair correlations of fractional powers of integers
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917927866859520 |
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| author | Sayous, Rafael |
| author_facet | Sayous, Rafael |
| contents | We study the statistics of pairs from the sequence $(n^α)_{n\in\mathbb{N}^*}$, for every parameter $α\in \, ]0,1[$. We prove the convergence of the empirical pair correlation measures towards a measure with an explicit density. In particular, when using the scaling factor $N\mapsto N^{1-α}$, we prove that there exists an exotic pair correlation function which exhibits a level repulsion phenomenon. For other scaling factors, we prove that either the pair correlations are Poissonian or there is a total loss of mass. In addition, we give an error term for this convergence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00793 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Effective pair correlations of fractional powers of integers Sayous, Rafael Number Theory Functional Analysis 11K38, 11J83, 28A33 We study the statistics of pairs from the sequence $(n^α)_{n\in\mathbb{N}^*}$, for every parameter $α\in \, ]0,1[$. We prove the convergence of the empirical pair correlation measures towards a measure with an explicit density. In particular, when using the scaling factor $N\mapsto N^{1-α}$, we prove that there exists an exotic pair correlation function which exhibits a level repulsion phenomenon. For other scaling factors, we prove that either the pair correlations are Poissonian or there is a total loss of mass. In addition, we give an error term for this convergence. |
| title | Effective pair correlations of fractional powers of integers |
| topic | Number Theory Functional Analysis 11K38, 11J83, 28A33 |
| url | https://arxiv.org/abs/2306.00793 |