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On Average Distance, Level-1 Fourier Weight, and Chang's Lemma

Lei Yu

TL;DR

This work advances the understanding of level-1 Fourier weights by deriving two new inductive bounds that improve Chang's lemma across the full range of densities $a\in[0,\tfrac12]$, including a small-$a$ bound with a carefully chosen threshold $T=0.21$ and a large-$a$ bound that is asymptotically tight as $a\to\tfrac12$. It also reframes the optimization of $W_1$ as an average-distance problem and applies these insights to strengthen the Friedgut–Kalai–Naor theorem for balanced functions, while establishing a sharp version of Chang's lemma for $\mathbb{F}_{2}^{n}$ by showing Hamming balls optimize the dimension of the spectrum spanned by large Fourier coefficients. The results connect extremal Fourier structure to geometric extremizers (linear threshold sets and Euclidean balls), offering precise bounds and asymptotic tightness in multiple regimes and linking discrete and Euclidean geometric intuitions. Collectively, the contributions yield tighter, regime-appropriate bounds with broader implications for additive combinatorics, Fourier analysis on the hypercube, and threshold-function extremality, and they point to concrete open problems such as the exact value of $W(1/8)$ and the precise convergence rate in $W(\tfrac12,\beta)$.

Abstract

In this paper, we improve the well-known level-1 weight bound, also known as Chang's lemma, by using an induction method. Our bounds are close to optimal no matter when the set is large or small. Our bounds can be seen as bounds on the minimum average distance problem, since maximizing the level-1 weight is equivalent to minimizing the average distance. We apply our new bounds to improve the Friedgut--Kalai--Naor theorem. We also derive the sharp version for Chang's original lemma for $\mathbb{F}_{2}^{n}$. That is, we show that in $\mathbb{F}_{2}^{n}$, Hamming balls maximize the dimension of the space spanned by large Fourier coefficients.

On Average Distance, Level-1 Fourier Weight, and Chang's Lemma

TL;DR

This work advances the understanding of level-1 Fourier weights by deriving two new inductive bounds that improve Chang's lemma across the full range of densities , including a small- bound with a carefully chosen threshold and a large- bound that is asymptotically tight as . It also reframes the optimization of as an average-distance problem and applies these insights to strengthen the Friedgut–Kalai–Naor theorem for balanced functions, while establishing a sharp version of Chang's lemma for by showing Hamming balls optimize the dimension of the spectrum spanned by large Fourier coefficients. The results connect extremal Fourier structure to geometric extremizers (linear threshold sets and Euclidean balls), offering precise bounds and asymptotic tightness in multiple regimes and linking discrete and Euclidean geometric intuitions. Collectively, the contributions yield tighter, regime-appropriate bounds with broader implications for additive combinatorics, Fourier analysis on the hypercube, and threshold-function extremality, and they point to concrete open problems such as the exact value of and the precise convergence rate in .

Abstract

In this paper, we improve the well-known level-1 weight bound, also known as Chang's lemma, by using an induction method. Our bounds are close to optimal no matter when the set is large or small. Our bounds can be seen as bounds on the minimum average distance problem, since maximizing the level-1 weight is equivalent to minimizing the average distance. We apply our new bounds to improve the Friedgut--Kalai--Naor theorem. We also derive the sharp version for Chang's original lemma for . That is, we show that in , Hamming balls maximize the dimension of the space spanned by large Fourier coefficients.

Paper Structure

This paper contains 14 sections, 16 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Preliminaries
  4. Organization
  5. Improved Bounds on Level-$1$ Weight
  6. Improved Bound for $a<1/4$
  7. Improved Bound for $a>1/4$
  8. Comparisons
  9. Application to Friedgut--Kalai--Naor Theorem
  10. Average Distance in Euclidean Spaces
  11. Chang's Original Lemma for $\mathbb{F}_{2}^{n}$
  12. Open Problems
  13. Alternative Proof of Theorem \ref{['thm:optimality']}
  14. Proof of Lemma \ref{['lem:Gamma']}

Key Result

Theorem 1

Let $f:\{\pm1\}^{n}\to\{0,1\}$ have expectation $a\in(0,1/2]$. Then,

Figures (1)

  • Figure 1: Comparisons of Chang's bound in Theorem \ref{['thm:level-1-bound']}, the bounds in Theorem \ref{['thm:bound']} and \ref{['thm:bound2']}, $\mathbf{W}_{1}$ of subcubes, and $\mathbf{W}_{1}$ of Hamming balls. In the left figure, the solid curves from top to bottom are respectively Chang's bound in Theorem \ref{['thm:level-1-bound']}, the minimum of the bounds in Theorem \ref{['thm:bound']} and \ref{['thm:bound2']}, and $\mathbf{W}_{1}$ of Hamming balls. The red squares are $\mathbf{W}_{1}$ of subcubes.

Theorems & Definitions (16)

  • Theorem 1: Level-1 Weight Bound
  • Theorem 2: Self-Consistency
  • Corollary 1
  • Theorem 3: Strong Level-1 Weight Bound
  • Proposition 1
  • Proposition 2
  • Theorem 4
  • Lemma 1
  • Theorem 5
  • Lemma 2
  • ...and 6 more