On the determinants of matrices with elements from arbitrary sets
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910560411451392 |
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| author | Shkredov, Ilya D. Shparlinski, Igor E. |
| author_facet | Shkredov, Ilya D. Shparlinski, Igor E. |
| contents | Recently there has been several works estimating the number of $n\times n$ matrices with elements from some finite sets $\mathcal X$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial upper bound $O(X^{n^2-1})$, where $X$ is the cardinality of $\mathcal X$. Here we show that even for arbitrary sets $\mathcal X\subseteq \mathbb R$,some recent results from additive combinatorics enable us to obtain a stronger bound with a power saving. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_04350 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the determinants of matrices with elements from arbitrary sets Shkredov, Ilya D. Shparlinski, Igor E. Number Theory Combinatorics Recently there has been several works estimating the number of $n\times n$ matrices with elements from some finite sets $\mathcal X$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial upper bound $O(X^{n^2-1})$, where $X$ is the cardinality of $\mathcal X$. Here we show that even for arbitrary sets $\mathcal X\subseteq \mathbb R$,some recent results from additive combinatorics enable us to obtain a stronger bound with a power saving. |
| title | On the determinants of matrices with elements from arbitrary sets |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2408.04350 |