On commuting integer matrices
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908332116148224 |
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| author | Chapman, Jonathan Mudgal, Akshat |
| author_facet | Chapman, Jonathan Mudgal, Akshat |
| contents | Given $d, N \in \mathbb{N}$, we define $\mathfrak{C}_d(N)$ to be the number of pairs of $d\times d$ matrices $A,B$ with entries in $[-N,N] \cap \mathbb{Z}$ such that $AB = BA$. We prove that $$ N^{10} \ll \mathfrak{C}_3(N) \ll N^{10},$$ thus confirming a speculation of Browning-Sawin-Wang. We further establish that $$ \mathfrak{C}_2(N) = K(2N+1)^5 (1 + o(1)),$$ where $K>0$ is an explicit constant. Our methods are completely elementary and rely on upper bounds of the correct order for restricted divisor correlations with high uniformity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_15839 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On commuting integer matrices Chapman, Jonathan Mudgal, Akshat Number Theory Combinatorics 11D45 (Primary), 15A27, 15B36 (Secondary) Given $d, N \in \mathbb{N}$, we define $\mathfrak{C}_d(N)$ to be the number of pairs of $d\times d$ matrices $A,B$ with entries in $[-N,N] \cap \mathbb{Z}$ such that $AB = BA$. We prove that $$ N^{10} \ll \mathfrak{C}_3(N) \ll N^{10},$$ thus confirming a speculation of Browning-Sawin-Wang. We further establish that $$ \mathfrak{C}_2(N) = K(2N+1)^5 (1 + o(1)),$$ where $K>0$ is an explicit constant. Our methods are completely elementary and rely on upper bounds of the correct order for restricted divisor correlations with high uniformity. |
| title | On commuting integer matrices |
| topic | Number Theory Combinatorics 11D45 (Primary), 15A27, 15B36 (Secondary) |
| url | https://arxiv.org/abs/2504.15839 |