An uncertainty principle for cyclic groups of prime order

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 $\mathbb{Z}/p\mathbb{Z}$ 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 $\mathbb{Z}/p\mathbb{Z}$ consisting of $k+1$ monomials can have at most $k$ zeroes. Another consequence is a short proof of the well-known Cauchy-Davenport inequality.

7 pages, no figures, submitted, Math Research Letters. More references added

References in corpus (1)

• Simple proof of Chebotarev's theorem on roots of unity

Cited by in corpus (1)

• Discrete Sampling and Interpolation: Universal Sampling Sets for Discrete Bandlimited Spaces