Gauss sums over some matrix groups

TL;DR

This paper derives explicit formulas for Gauss sums over certain matrix groups over finite fields, using averaging techniques, and applies these results to matrix counting problems and bounds in additive combinatorics.

Contribution

It provides new explicit expressions for Gauss sums over linear groups and applies these to matrix enumeration and bound improvements.

Findings

Explicit formulas for Gauss sums over linear groups

Counting invertible matrices with zero-trace over finite fields

Improved bounds in additive combinatorics problems

Abstract

In this note, we give explicit expressions of Gauss sums for general (resp. special) linear groups over finite fields, which involves Gauss sums (resp. Kloosterman sums). The key ingredient is averaging such sums over Borel subgroups. As applications, we count the number of invertible matrices of zero-trace over finite fields and we also improve two bounds by Ferguson, Hoffman, Luca, Ostafe and Shparlinski in [ Some additive combinatorics problems in matrix rings, Rev. Mat. Complut. (23) 2010, 501--513 ].