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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.04814 |
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| _version_ | 1866913493230288896 |
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| author | Gath, Yoav A. |
| author_facet | Gath, Yoav A. |
| contents | Let $E(x;ω)$ be the error term for the number of integer lattice points lying inside a $3$-dimensional Cygan-Korányi spherical shell of inner radius $x$ and gap width $ω(x)>0$. Assuming that $ω(x)\to0$ as $x\to\infty$, and that $ω$ satisfies suitable regularity conditions, we prove that $E(x;ω)$, properly normalized, has a limiting distribution. Moreover, we show that the corresponding distribution is moment-determinate, and we give a closed form expression for its moments. As a corollary, we deduce that the limiting distribution is the standard Gaussian measure whenever $ω$ is slowly varying. We also construct gap width functions $ω$, whose corresponding error term has a limiting distribution that is absolutely continuous with a non-Gaussian density. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04814 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lattice point counting statistics for 3-dimensional shrinking Cygan-Korányi spherical shells Gath, Yoav A. Number Theory Let $E(x;ω)$ be the error term for the number of integer lattice points lying inside a $3$-dimensional Cygan-Korányi spherical shell of inner radius $x$ and gap width $ω(x)>0$. Assuming that $ω(x)\to0$ as $x\to\infty$, and that $ω$ satisfies suitable regularity conditions, we prove that $E(x;ω)$, properly normalized, has a limiting distribution. Moreover, we show that the corresponding distribution is moment-determinate, and we give a closed form expression for its moments. As a corollary, we deduce that the limiting distribution is the standard Gaussian measure whenever $ω$ is slowly varying. We also construct gap width functions $ω$, whose corresponding error term has a limiting distribution that is absolutely continuous with a non-Gaussian density. |
| title | Lattice point counting statistics for 3-dimensional shrinking Cygan-Korányi spherical shells |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.04814 |