Matrix Kloosterman sums

TL;DR

This paper investigates a family of exponential sums related to the horocyclic flow on GL_n, establishing explicit purity results and optimal bounds to deepen understanding of their mathematical properties.

Contribution

It provides an explicit version of general purity and determines optimal bounds for matrix Kloosterman sums, advancing the theoretical understanding of these sums.

Findings

Proved explicit purity for matrix Kloosterman sums.

Established optimal bounds for these exponential sums.

Enhanced understanding of sums related to horocyclic flow on GL_n.

Abstract

We study a family of exponential sums that arises in the study of the horocyclic flow on $\mathrm{GL}_n$. We prove an explicit version of general purity and find optimal bounds for these sums.