Bounds for Kloosterman Sums for $\mathrm{GL}_n$

TL;DR

This paper establishes improved power-saving bounds for Kloosterman sums on $ ext{GL}_n$, extending previous trivial bounds by explicitly representing sums as exponential sums and applying Weil bounds.

Contribution

It provides the first non-trivial bounds for Kloosterman sums on $ ext{GL}_n$ for all Weyl elements, generalizing prior results for $n>2$.

Findings

Improved bounds for Kloosterman sums on $ ext{GL}_n$

Explicit representation of sums as exponential sums

Application of Weil bounds to obtain power savings

Abstract

In this paper power saving bounds for general Kloosterman sums for all Weyl elements for $\mathrm{GL}_n$ for $n>2$ are proven, improving the trivial bound by D\k{a}browski and Reeder. This is achieved by representing the sums in an explicit way as exponential sums and bounding these through applications of the Weil bound.