Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor - Gorodetsky, O; Kovaleva, V (Wiley, 2024)
Let g be a random matrix distributed according to uniform probability measure on the finite general linear group GLn⁢(Fq). We show that Tr⁡(gk) equidistributes on Fq as n→∞ as long as log⁡k=o⁡(n2) and that this range is sharp. We also show that nontrivial linear combinations of Tr⁡(g1),…,Tr⁡(gk) 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⩽cq⁢n, where cq 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 Tr⁡(gk), we end up showing that certain explicit character sums modulo Tk+1 exhibit cancellation when averaged over monic polynomials of degree n in Fq⁡[T] as long as log⁡k=o⁡(n2). 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.