Inequalities in Fourier analysis on binary cubes

TL;DR

This paper precisely characterizes the ranges of Lebesgue exponents for which sharp Fourier inequalities hold on binary cubes, leading to new bounds on additive energies, entropic uncertainty, and Fourier restriction in this discrete setting.

Contribution

It provides the exact ranges of exponents for sharp inequalities on binary cubes and derives several new bounds and principles in this discrete Fourier analysis setting.

Findings

Sharp bounds on generalized additive energies of subsets of binary cubes.

A binary variant of the Beckner-Hirschman entropic uncertainty principle.

Exact dimension-free estimates for Fourier restriction to the binary cube.

Abstract

This paper studies two classical inequalities, namely the Hausdorff-Young inequality and equal-exponent Young's convolution inequality, for discrete functions supported in the binary cube $\{0,1\}^d\subset\mathbb{Z}^d$. We characterize the exact ranges of Lebesgue exponents in which sharp versions of these two inequalities hold, and present several immediate consequences. First, if the functions are specialized to be the indicator of some set $A\subseteq\{0,1\}^d$, then we obtain sharp upper bounds on two types of generalized additive energies of $A$, extending the works of Kane-Tao, de Dios Pont-Greenfeld-Ivanisvili-Madrid, and one of the present authors. Second, we obtain a sharp binary variant of the Beckner-Hirschman entropic uncertainty principle, as well as a sharp lower estimate on the entropy of a sum of two independent random variables with values in $\{0,1\}^d$. Finally, the sharp binary Hausdorff-Young inequality also reveals the exact range of dimension-free estimates for the Fourier restriction to the binary cube.