Average variance bounds for integer points on the sphere
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929248745291776 |
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| author | Lutsko, Christopher |
| author_facet | Lutsko, Christopher |
| contents | Consider the integer points lying on the sphere of fixed radius projected onto the unit sphere. Duke showed that, on congruence conditions for the radius squared, these points equidistribute. To further this study of equidistribution, we consider the variance of the number of points in a spherical cap. An asymptotic for this variance was conjectured by Bourgain-Rudnick-Sarnak. We prove an upper bound of the correct size on the average (over radii) of these variances. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12822 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Average variance bounds for integer points on the sphere Lutsko, Christopher Number Theory 11E16, 11F30 Consider the integer points lying on the sphere of fixed radius projected onto the unit sphere. Duke showed that, on congruence conditions for the radius squared, these points equidistribute. To further this study of equidistribution, we consider the variance of the number of points in a spherical cap. An asymptotic for this variance was conjectured by Bourgain-Rudnick-Sarnak. We prove an upper bound of the correct size on the average (over radii) of these variances. |
| title | Average variance bounds for integer points on the sphere |
| topic | Number Theory 11E16, 11F30 |
| url | https://arxiv.org/abs/2402.12822 |