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Khintchine inequalities and $Z_2$ sets

Chian Yeong Chuah, Zhen-Chuan Liu, Tao Mei

TL;DR

The paper connects noncommutative Khintchine inequalities at the critical $p=1$ constant $1/\sqrt{2}$ to $Z_2$-set structure, proving that any subset of a discrete group (in particular the integers) satisfying this optimal bound must be a $Z_2$-set. It develops a robust operator-valued framework using noncommutative $L_p$ spaces, column/row spaces, and group von Neumann algebras to establish a two-sided inequality controlled by a computable $\alpha(W)$ from the $Z_2$ data. The main result yields concrete corollaries for specific sequences (Haar unitaries, Gaussian, Chebyshev) and provides partial converses, giving a deep link between harmonic-analysis properties of subsets and probabilistic Khintchine-type bounds. The work extends to broad classes of discrete groups and clarifies the role of the $Z_2$ constant in governing optimal constants, with implications for Sidon-type phenomena and abelian group settings.

Abstract

In this paper we show that the subset of integers that satisfies the Khintchine inequality for $p=1$ with the optimal constant ${\sqrt{2}}$ has to be a $Z_2$ set. We further prove a similar result for a large class of discrete groups. Our arguments rely on previous works by Haagerup/Musat \cite{Haagerup2007}, and Haagerup/Itoh \cite{Haagerup1995}.

Khintchine inequalities and $Z_2$ sets

TL;DR

The paper connects noncommutative Khintchine inequalities at the critical constant to -set structure, proving that any subset of a discrete group (in particular the integers) satisfying this optimal bound must be a -set. It develops a robust operator-valued framework using noncommutative spaces, column/row spaces, and group von Neumann algebras to establish a two-sided inequality controlled by a computable from the data. The main result yields concrete corollaries for specific sequences (Haar unitaries, Gaussian, Chebyshev) and provides partial converses, giving a deep link between harmonic-analysis properties of subsets and probabilistic Khintchine-type bounds. The work extends to broad classes of discrete groups and clarifies the role of the constant in governing optimal constants, with implications for Sidon-type phenomena and abelian group settings.

Abstract

In this paper we show that the subset of integers that satisfies the Khintchine inequality for with the optimal constant has to be a set. We further prove a similar result for a large class of discrete groups. Our arguments rely on previous works by Haagerup/Musat \cite{Haagerup2007}, and Haagerup/Itoh \cite{Haagerup1995}.

Paper Structure

This paper contains 12 sections, 15 theorems, 76 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Noncommutative $L_p$ spaces and orthonormal system
  4. Group von Neuman algebra
  5. $Z_2$-sets
  6. Column and row spaces
  7. Noncommutative-Khintchine inequality
  8. Main results and proof
  9. Statement of the main theorem
  10. Proof of the main theorem
  11. Some converse results
  12. Appendix

Key Result

Theorem 1.0.1

Let $d,n\in \mathbb{N}$ and $C_1,\cdots,C_d\in M_n(\mathbb{C})$, then

Figures (2)

  • Figure 1: The case when $d_i\neq 0, i=1,\cdots,6$
  • Figure 3: The case when $d_3\neq 0, d_5\neq 0$ and all else are 0

Theorems & Definitions (37)

  • Theorem 1.0.1: LustPiquard1991,Haagerup2007
  • Definition 2.1.1
  • Remark 2.2.1
  • Definition 2.3.1
  • Remark 2.3.2
  • Example 2.3.3
  • Remark 2.3.4
  • Definition 2.5.1
  • Theorem 3.1.1
  • Example 3.1.2
  • ...and 27 more