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Fourier inequalities and sign uncertainty

Roni Edwin

TL;DR

The paper investigates the sign uncertainty principle in high dimensions and establishes a stronger asymptotic lower bound for the ±1 sign-uncertainty constants. The central technique proves that for any $f$ in $H^d$, $||f||_{L^2}^2 <= (2/e)^{d/2} ||f||_{L^1} ||f_hat||_{L^1}$, which in turn implies $A_s(d)/sqrt{d} >= 1/(2 sqrt(pi))$ for $d \ge 5$ and furnishes improved lower bounds for the Cohn-Elkies linear program used in sphere packing. The work also shows that this inequality yields a lower bound on Delta_d^{LP} of at least $(1/4) (e/8)^{d/2}$, aligning with Torquato-Stillinger predictions. It discusses the potential for tightening constants via radial symmetry and notes connections to extremisers and the asymptotic behavior of $A_+(d)$ and $A_-(d)$.

Abstract

Motivated by inequalities in Fourier analysis, we present an improvement on the lower bound for the sign uncertainty principle of Bourgain, Clozel and Kahane in high dimensions. Additionally, our methods can be used to match the existing Torquato-Stillinger lower bounds for the Cohn-Elkies linear program for sphere packing.

Fourier inequalities and sign uncertainty

TL;DR

The paper investigates the sign uncertainty principle in high dimensions and establishes a stronger asymptotic lower bound for the ±1 sign-uncertainty constants. The central technique proves that for any in , , which in turn implies for and furnishes improved lower bounds for the Cohn-Elkies linear program used in sphere packing. The work also shows that this inequality yields a lower bound on Delta_d^{LP} of at least , aligning with Torquato-Stillinger predictions. It discusses the potential for tightening constants via radial symmetry and notes connections to extremisers and the asymptotic behavior of and .

Abstract

Motivated by inequalities in Fourier analysis, we present an improvement on the lower bound for the sign uncertainty principle of Bourgain, Clozel and Kahane in high dimensions. Additionally, our methods can be used to match the existing Torquato-Stillinger lower bounds for the Cohn-Elkies linear program for sphere packing.

Paper Structure

This paper contains 2 sections, 5 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['mainthm1']}

Key Result

Theorem 1

Suppose $f\colon \mathbb{R} \to \mathbb{C}$ is a measurable function such that $|f(x)|\le Ce^{- a\pi x^2}$ and $\bigl|\widehat{f}(\omega)\bigr|\le C'e^{-\frac{\pi\omega^2}{a} }$ for all $x,\omega \in \mathbb{R}$. Then $f$ is a scalar multiple of the Gaussian $x \mapsto e^{-a \pi x^2}$.

Theorems & Definitions (7)

  • Theorem : Hardy uncertainty principle, hardy1933theorem
  • Theorem 1.1: bourgain2010principe, Théorème $1$
  • Theorem 1.4
  • Theorem 1.5
  • proof : Proof of Theorem \ref{['bodya']} under Theorem \ref{['mainthm1']}
  • Theorem 1.6: Theorem 3.2 in cohn2003new, Theorem 3.3 in Cohn_2014
  • Remark