An uncertainty principle for cyclic groups of prime order
| dc.creator | Tao, Terence | |
| dc.date | 2003-08-29 | |
| dc.date | 2004-07-22 | |
| dc.date.accessioned | 2026-07-07T05:00:41Z | |
| dc.date.available | 2026-07-07T05:00:41Z | |
| dc.description | 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. | |
| dc.description | 7 pages, no figures, submitted, Math Research Letters. More references added | |
| dc.identifier | https://arxiv.org/abs/math/0308286 | |
| dc.identifier | http://arxiv.org/abs/math/0308286 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/68409 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Number Theory | |
| dc.subject | 42A99 | |
| dc.title | An uncertainty principle for cyclic groups of prime order | |
| dc.type | text |