Inequalities in Fourier analysis on binary cubes

2025-07-02

AI Summary

This paper precisely determines the sharp Lebesgue exponents for the Hausdorff–Young inequality and the endpoint Young convolution inequality in discrete Fourier analysis on the binary cube ${0,1}^d$. Employing a unified framework integrating harmonic analysis (discrete Fourier transform), extremal combinatorics, and information theory (entropy estimation), the authors develop a novel analytical toolkit for high-dimensional Boolean spaces. Key contributions include: (i) the first exact characterization of the optimal exponent ranges for both inequalities; (ii) a tight upper bound on the generalized additive energy of set indicator functions; (iii) the formulation and proof of a sharp binary entropy uncertainty principle; and (iv) dimension-independent Fourier restriction-type estimates. These results establish foundational tools with broad implications for additive combinatorics, information theory, and discrete signal processing.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Machine Learning: Information Theory
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Content-based information diffusion

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 ${0,1}^dsubsetmathbb{Z}^d$. 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 $Asubseteq{0,1}^d$, 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.

Problem

• Characterize sharp Hausdorff-Young inequality ranges on binary cubes
• Determine sharp Young's convolution inequality for discrete functions
• Establish dimension-free Fourier restriction bounds for binary cubes

Innovation

• Characterizes sharp Lebesgue exponent ranges
• Extends generalized additive energy bounds
• Develops sharp binary entropic uncertainty principles