Mathematical Research Letters

Volume 12 (2005)

Number 1

An uncertainty principle for cyclic groups of prime order

Pages: 121 – 127

DOI: http://dx.doi.org/10.4310/MRL.2005.v12.n1.a11

Author

Terence Tao (University of California at Los Angeles)

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.

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