An uncertainty inequality for finite abelian groups

Let G be a finite abelian group of order n. For a complex valued function f on G let f̂ denote the Fourier transform of f . The classical uncertainty inequality asserts that if f 6= 0 then |supp(f)| · |supp(f̂)| ≥ |G| . (1) Answering a question of Terence Tao, the following improvement of (1) is shown: Theorem: Let d1 < d2 be two consecutive divisors of n. If d1 ≤ k = |supp(f)| ≤ d2 then |supp(f̂)| ≥ n d1d2 (d1 + d2 − k) .