An uncertainty inequality for finite abelian groups

Abstract

Let G be a finite abelian group of order n. For a complex valued function f on G, let \fht denote the Fourier transform of f. The uncertainty inequality asserts that if f \neq 0 then |supp(f)| |supp(\fht)| \geq n. Answering a question of Terence Tao, the following improvement of the classical inequality is shown: Let d_1<d_2 be two consecutive divisors of n. If d_1 \leq k=|supp(f)| \leq d_2 then: |supp(\fht)| \geq \frac{n(d_1+d_2-k)}{d_1 d_2}