Estimation of function's supports under arithmetic constraints

Abstract

The well-known $|supp(f)||supp(\widehat{f}|\geq |G|$ inequality gives lower estimation of each supports. In the present paper we give upper estimation under arithmetic constrains. The main notion will be the additive energy which plays a central role in additive combinatorics. We prove an uncertainty inequality that shows a trade-off between the total changes of the indicator function of a subset $A\subseteq \mathbb F^n_2$ and the additive energy of $A$ and the Fourier spectrum.