Poissonian pair correlation for higher dimensional real sequences
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929433875578880 |
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| author | Bera, Tanmoy Das, Mithun Kumar Mukhopadhyay, Anirban |
| author_facet | Bera, Tanmoy Das, Mithun Kumar Mukhopadhyay, Anirban |
| contents | In this article, we examine the Poissonian pair correlation (PPC) statistic for higher-dimensional real sequences. Specifically, we demonstrate that for $d\geq 3$, almost all $(α_1,\ldots,α_d) \in \mathbb{R}^d$, the sequence $\big(\{x_nα_1\},\dots,\{x_nα_d\}\big)$ in $[0,1)^d$ has PPC conditionally on the additive energy bound of $(x_n).$ This bound is more relaxed compared to the additive energy bound for one dimension as discussed in [1]. More generally, we derive the PPC for $\big(\{x_n^{(1)}α_1\},\dots,\{x_n^{(d)}α_d\}\big) \in [0,1)^d$ for almost all $(α_1,\ldots,α_d) \in \mathbb{R}^d.$ As a consequence we establish the metric PPC for $(n^{θ_1},\ldots,n^{θ_d})$ provided that all of the $θ_i$'s are greater than two. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_09541 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Poissonian pair correlation for higher dimensional real sequences Bera, Tanmoy Das, Mithun Kumar Mukhopadhyay, Anirban Number Theory 11K06, 11J83, 11M06, 11J25, 11J71, 42B05, 11L07 In this article, we examine the Poissonian pair correlation (PPC) statistic for higher-dimensional real sequences. Specifically, we demonstrate that for $d\geq 3$, almost all $(α_1,\ldots,α_d) \in \mathbb{R}^d$, the sequence $\big(\{x_nα_1\},\dots,\{x_nα_d\}\big)$ in $[0,1)^d$ has PPC conditionally on the additive energy bound of $(x_n).$ This bound is more relaxed compared to the additive energy bound for one dimension as discussed in [1]. More generally, we derive the PPC for $\big(\{x_n^{(1)}α_1\},\dots,\{x_n^{(d)}α_d\}\big) \in [0,1)^d$ for almost all $(α_1,\ldots,α_d) \in \mathbb{R}^d.$ As a consequence we establish the metric PPC for $(n^{θ_1},\ldots,n^{θ_d})$ provided that all of the $θ_i$'s are greater than two. |
| title | Poissonian pair correlation for higher dimensional real sequences |
| topic | Number Theory 11K06, 11J83, 11M06, 11J25, 11J71, 42B05, 11L07 |
| url | https://arxiv.org/abs/2310.09541 |