An Uncertainty Principle for Cyclic Groups of Prime Order
نویسنده
چکیده
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) 6= 0} is related to the support of the Fourier transform f̂ : G → C by the formula |supp(f)||supp(f̂)| ≥ |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(f̂)| ≥ p + 1 and show that this is absolutely sharp. As one consequence, we see that a sparse polynomial in Z/pZ consisting of k monomials can have at most k zeroes. Another consequence is a short proof of the well-known CauchyDavenport inequality.
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