Abstract
We establish a number of uncertainty inequalities for the additive group of a finite affine plane, showing that for p prime, a nonzero function f:Fp2→C and its Fourier transform fˆ:Fp2ˆ→C cannot have small supports simultaneously. The “baseline” of our investigation is the well-known Meshulam's bound, which we sharpen, for the particular groups under consideration, taking into account not only the sizes of the support sets supp f and suppfˆ, but also their structure.Our results imply in particular that, with some explicitly classified exceptions, one has |suppf||suppfˆ|≥3p(p−2); in comparison, the classical uncertainty inequality gives |suppf||suppfˆ|≥p2.
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