Distribution of integer points on determinant surfaces and a $\text{mod-}p$ analogue
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| Format: | Preprint |
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2025
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| _version_ | 1866914605921468416 |
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| author | Ganguly, Satadal Guria, Rachita |
| author_facet | Ganguly, Satadal Guria, Rachita |
| contents | We establish an asymptotic formula for counting integer solutions with smooth weights to an equation of the form $xy-zw=r$, where $r$ is a non-zero integer, with an explicit main term and a strong bound on the error term in terms of the size of the variables $x, y, z, w$ as well as of $r$. We also establish an asymptotic formula for counting integer solutions with smooth weights to the congruence $xy-zw \equiv 1 (\text{mod }p)$, where $p$ is a large prime, with a strong bound on the error term. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_14793 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distribution of integer points on determinant surfaces and a $\text{mod-}p$ analogue Ganguly, Satadal Guria, Rachita Number Theory 11 E20, 11E25, 11F30, 11F72, 11N45 We establish an asymptotic formula for counting integer solutions with smooth weights to an equation of the form $xy-zw=r$, where $r$ is a non-zero integer, with an explicit main term and a strong bound on the error term in terms of the size of the variables $x, y, z, w$ as well as of $r$. We also establish an asymptotic formula for counting integer solutions with smooth weights to the congruence $xy-zw \equiv 1 (\text{mod }p)$, where $p$ is a large prime, with a strong bound on the error term. |
| title | Distribution of integer points on determinant surfaces and a $\text{mod-}p$ analogue |
| topic | Number Theory 11 E20, 11E25, 11F30, 11F72, 11N45 |
| url | https://arxiv.org/abs/2508.14793 |