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 f ̂ denote the Fourier transform of f . The classical uncertainty inequality asserts that if f ≠ 0 then (1) | supp ( f ) | ⋅ | supp ( f ̂ ) | ≥ | G | . Answering a question of Terence Tao, the following improvement of (1) is shown: m. Let d 1 < d 2 be two consecutive divisors of n . If d 1 ≤ k = | supp ( f ) | ≤ d 2 then | supp ( f ̂ ) | ≥ n d 1 d 2 ( d 1 + d 2 − k ) .