An uncertainty principle for cyclic groups of prime order

TL;DR

This paper establishes a sharper uncertainty principle for functions on cyclic groups of prime order, linking support sizes of functions and their Fourier transforms, with implications for sparse polynomials and additive number theory.

Contribution

It proves an improved support inequality for functions on cyclic groups of prime order, strengthening the classical uncertainty principle.

Findings

Supports the inequality | ext{supp}(f)| + | ext{supp}( ext{hat}f)| \, ext{geq} \, p+1

Shows that a sparse polynomial with k+1 monomials has at most k zeros

Provides a short proof of the Cauchy-Davenport inequality

Abstract

Let $G$ be a finite abelian group, and let $f: G \to \C$ be a complex function on $G$. The uncertainty principle asserts that the support $\supp(f) := \{x \in G: f(x) \neq 0\}$ is related to the support of the Fourier transform $\hat f: G \to \C$ by the formula $$ |\supp(f)| |\supp(\hat f)| \geq |G|$$ where $|X|$ denotes the cardinality of $X$. In this note we show that when $G$ is the cyclic group $\Z/p\Z$ of prime order $p$, then we may improve this to $$ |\supp(f)| + |\supp(\hat f)| \geq p+1$$ and show that this is absolutely sharp. As one consequence, we see that a sparse polynomial in $\Z/p\Z$ consisting of $k+1$ monomials can have at most $k$ zeroes. Another consequence is a short proof of the well-known Cauchy-Davenport inequality.