Supercharacters, exponential sums, and the uncertainty principle

TL;DR

This paper explores supercharacter theories on finite abelian groups, linking them to exponential sums in number theory, and develops a generalized Fourier transform with an uncertainty principle.

Contribution

It introduces a framework connecting supercharacters to classical exponential sums and generalizes the Fourier transform with new uncertainty bounds.

Findings

Supercharacters induce classical exponential sums like Gauss and Kloosterman sums.

A generalized Fourier transform based on supercharacters is developed.

Uncertainty principles with explicit constants are established for this transform.

Abstract

The theory of supercharacters, which generalizes classical character theory, was recently introduced by P. Diaconis and I.M. Isaacs, building upon earlier work of C. Andre. We study supercharacter theories on $(Z/nZ)^d$ induced by the actions of certain matrix groups, demonstrating that a variety of exponential sums of interest in number theory (e.g., Gauss, Ramanujan, Heilbronn, and Kloosterman sums) arise in this manner. We develop a generalization of the discrete Fourier transform, in which supercharacters play the role of the Fourier exponential basis. We provide a corresponding uncertainty principle and compute the associated constants in several cases.