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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2310.19578 |
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| _version_ | 1866915056526032896 |
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| author | Sayous, Rafael |
| author_facet | Sayous, Rafael |
| contents | Using a standard definition of fractional powers on the universal cover $\exp:S\to \mathbb{C}^*$ seen as an infinite helicoid embedded in $\mathbb{R}^3$, we study the statistics of pairs from the countable family $\{n^α\, : \, n \in \exp^{-1}(Λ) \}$ for every complex grid $Λ$ and every real parameter $α\in \, ]0,1[\,$. We prove the convergence of the empirical pair correlations measures towards a rotation invariant measure with explicit density. In particular, with 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, with explicit dependence on parameters of the grid $Λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_19578 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Effective pair correlations of fractional powers of complex grid points Sayous, Rafael Number Theory 11J83, 11K38, 11P21, 28A33 Using a standard definition of fractional powers on the universal cover $\exp:S\to \mathbb{C}^*$ seen as an infinite helicoid embedded in $\mathbb{R}^3$, we study the statistics of pairs from the countable family $\{n^α\, : \, n \in \exp^{-1}(Λ) \}$ for every complex grid $Λ$ and every real parameter $α\in \, ]0,1[\,$. We prove the convergence of the empirical pair correlations measures towards a rotation invariant measure with explicit density. In particular, with 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, with explicit dependence on parameters of the grid $Λ$. |
| title | Effective pair correlations of fractional powers of complex grid points |
| topic | Number Theory 11J83, 11K38, 11P21, 28A33 |
| url | https://arxiv.org/abs/2310.19578 |