An uncertainty principle for cyclic groups of prime order

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