An improved uncertainty principle for functions with symmetry

TL;DR

This paper generalizes the uncertainty principle for functions with symmetries, extending Chebotar"ev's non-vanishing minors result to cosine and sine matrices, and explores implications over prime and non-prime fields.

Contribution

It introduces a generalized uncertainty principle for symmetric functions, including new results for discrete cosine and sine matrices, and demonstrates the optimality and strength of these results.

Findings

Nonzero minors for discrete Fourier, cosine, and sine matrices.

Generalization of the Biró-Meshulam-Tao uncertainty principle.

Results applicable over prime and certain non-prime fields.

Abstract

Chebotar\"ev proved that every minor of a discrete Fourier matrix of prime order is nonzero. We prove a generalization of this result that includes analogues for discrete cosine and discrete sine matrices as special cases. We establish these results via a generalization of the Bir\'o-Meshulam-Tao uncertainty principle to functions with symmetries that arise from certain group actions, with some of the simplest examples being even and odd functions. We show that our result is best possible and in some cases is stronger than that of Bir\'o-Meshulam-Tao. Some of these results hold in certain circumstances for non-prime fields; Gauss sums play a central role in such investigations.