Estimation of function's supports under arithmetic constraints

Abstract

The well-known inequality \(\lvert {\rm supp}(f) \rvert \lvert {\rm supp}( \widehat f) \rvert \geq |G|\) gives a lower estimation for each support. In this paper we consider the case where there exists a slowly increasing function \(F\) such that \(\lvert {\rm supp}(f) \rvert \leq F(\lvert {\rm supp}( \widehat f) \rvert )\). We will show that this can be done under some arithmetic constraint. The two links that help us come from additive combinatorics and theoretical computer science. The first is the additive energy which plays a central role in additive combinatorics. The second is the influence of Boolean functions. Our main tool is the spectral analysis of Boolean functions. We prove an uncertainty inequality in which the influence of a function and its Fourier spectrum play a role.