Inequalities in Fourier analysis on binary cubes

2025/07/02 by Tonći Crmarić, Vjekoslav Kovač, Crmarić, Tonći +3 · 1 citation
Engineering · Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Mathematical Approximation and Integration #Wireless Communication Security Techniques

paper · pdf · doi:10.48550/arxiv.2507.01359

openalex publication_date 2025/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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⊂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⊆\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.

Citations

• Dimension-free Fourier restriction inequalities
• Sharp estimates for Gowers norms on discrete cubes
• Discrete Brunn-Minkowski Inequality for subsets of the cube
• Additive energies on discrete cubes
• A Weighted Prékopa-Leindler inequality and sumsets with quasicubes
• A bound on partitioning clusters
• Energies and structure of additive sets
• Higher moments of convolutions
• A note about entropy

Cited by

• Functional Inequalities and Random Walks on Increasing Subsets of the Hypercube
• Optimal Young's convolutions inequality and its reverse form on the hypercube

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• Banach space valued Pisier and Riesz type inequalities on discrete cube