Estimation of function's supports under arithmetic constraints

The well-known $\mathrm{\mid }supp(f)\mathrm{\mid }\mathrm{\mid }supp(\widehat{f}\mathrm{\mid }\ge \mathrm{\mid }G\mathrm{\mid }$ 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 $A\subseteq {\mathbb{F}}_2^n$ and the additive energy of $A$ and the Fourier spectrum.