Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor
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arXiv
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| Format: | Preprint |
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2023
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| author | Gorodetsky, Ofir Kovaleva, Valeriya |
| author_facet | Gorodetsky, Ofir Kovaleva, Valeriya |
| contents | Let $g$ be a random matrix distributed according to uniform probability measure on the finite general linear group $\mathrm{GL}_n(\mathbb{F}_q)$. We show that $\mathrm{Tr}(g^k)$ equidistributes on $\mathbb{F}_q$ as $n \to \infty$ as long as $\log k=o(n^2)$ and that this range is sharp. We also show that nontrivial linear combinations of $\mathrm{Tr}(g^1),\ldots, \mathrm{Tr}(g^k)$ equidistribute as long as $\log k =o(n)$ and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for $k \le c_q n$, where $c_q$ depends on $q$, due to work of the first author and Rodgers.
We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of $\mathrm{Tr}(g^k)$ we end up showing that certain explicit character sums modulo $T^{k+1}$ exhibit cancellation when averaged over monic polynomials of degree $n$ in $\mathbb{F}_q[T]$ as long as $\log k = o(n^2)$. This goes far beyond the classical range $\log k =o(n)$ due to Montgomery and Vaughan. To study these sums we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_01344 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor Gorodetsky, Ofir Kovaleva, Valeriya Number Theory Probability Let $g$ be a random matrix distributed according to uniform probability measure on the finite general linear group $\mathrm{GL}_n(\mathbb{F}_q)$. We show that $\mathrm{Tr}(g^k)$ equidistributes on $\mathbb{F}_q$ as $n \to \infty$ as long as $\log k=o(n^2)$ and that this range is sharp. We also show that nontrivial linear combinations of $\mathrm{Tr}(g^1),\ldots, \mathrm{Tr}(g^k)$ equidistribute as long as $\log k =o(n)$ and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for $k \le c_q n$, where $c_q$ depends on $q$, due to work of the first author and Rodgers. We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of $\mathrm{Tr}(g^k)$ we end up showing that certain explicit character sums modulo $T^{k+1}$ exhibit cancellation when averaged over monic polynomials of degree $n$ in $\mathbb{F}_q[T]$ as long as $\log k = o(n^2)$. This goes far beyond the classical range $\log k =o(n)$ due to Montgomery and Vaughan. To study these sums we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums. |
| title | Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2307.01344 |