Equidistribution of Kloosterman sums over function fields

TL;DR

This paper establishes the Sato--Tate distribution for Kloosterman sums over function fields, providing explicit error bounds and extending results to joint distributions of different exponential sums.

Contribution

It proves the Sato--Tate distribution for Kloosterman sums over function fields with explicit error terms and extends to joint distributions of different sums.

Findings

Sato--Tate distribution of Kloosterman sums proven with explicit error bounds

Distribution results hold over arithmetic progressions and short intervals

Joint distribution of two different exponential sums established

Abstract

We prove the Sato--Tate distribution of Kloosterman sums over function fields with explicit error terms, when the places vary in arithmetic progressions or short intervals. A joint Sato--Tate distribution of two ``different" exponential sums is also proved.