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) over cap denote the Fourier transform of f. The classical uncertainty inequality asserts that if f not equal 0 then vertical bar supp(f)vertical bar center dot vertical bar supp((f) over cap)vertical bar >= vertical bar G vertical bar. (1) Answering a question of Terence Tao, the following improvement of (1) is shown: Theorem. Let d(1) < d(2) be two consecutive divisors of n. If d(1) <= k = vertical bar supp(f)vertical bar <= d(2) then vertical bar supp((f) over cap)vertical bar >= (d1d2)/(n)(d(1) + d(2) - k). (c) 2004 Elsevier Ltd. All rights reserved.