Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor

TL;DR

This paper proves that high traces of random matrices over finite fields become uniformly distributed as matrix size grows, extending previous results and connecting the problem to cancellation in character sums over function fields.

Contribution

It establishes sharp bounds for the equidistribution of traces and their linear combinations, extending the known range from linear to quadratic in the matrix size, and links the problem to character sum cancellation.

Findings

High traces equidistribute as matrix size increases, for log k=o(n^2)

Linear combinations of traces also equidistribute for log k=o(n)

Connection established between trace distribution and cancellation in short character sums

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.