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Mathematics > Number Theory
Title: Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor
(Submitted on 3 Jul 2023 (v1), last revised 8 May 2024 (this version, v2))
Abstract: 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.
Submission history
From: Ofir Gorodetsky [view email][v1] Mon, 3 Jul 2023 20:28:57 GMT (19kb)
[v2] Wed, 8 May 2024 08:46:55 GMT (20kb)
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