An uncertainty inequality for finite abelian groups

Let GG be a finite abelian group of order nn. For a complex valued function ff on GG let f̂ denote the Fourier transform of ff. The classical uncertainty inequality asserts that if f≠0f≠0 then equation(1)|supp(f)|⋅|supp(f̂)|≥|G|. Answering a question of Terence Tao, the following improvement of (1) is shown:Theorem. Let d1