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Remarks on the inverse Littlewood conjecture

Thomas F. Bloom, Ben Green

Abstract

The Littlewood conjecture, proven by Konyagin and McGehee-Pigno-Smith in the 1980s, states that if $A\subset \mathbb{Z}$ is a finite set of integers with $\lvert A\rvert=N$ then $\| \widehat{1_A}\|_1\geq c\log N$ for some absolute constant $c > 0$. We explore what structure $A$ must have if $\| \widehat{1_A}\|_1\leq K\log N$ for some constant $K$. Under such an assumption we prove, for instance, that $A$ contains a subset $A'\subseteq A$ with $\lvert A\rvert \geq N^{0.99}$ such that $\lvert A'+A'\rvert \ll K^{O(1)}\lvert A'\rvert$. As a consequence, for any $k\geq 3$, if $N$ is sufficiently large depending on $k$ and $K$, then $A$ must contain an arithmetic progression of length $k$. A byproduct of our analysis is a (slightly) improved bound for the constant $c$.

Remarks on the inverse Littlewood conjecture

Abstract

The Littlewood conjecture, proven by Konyagin and McGehee-Pigno-Smith in the 1980s, states that if is a finite set of integers with then for some absolute constant . We explore what structure must have if for some constant . Under such an assumption we prove, for instance, that contains a subset with such that . As a consequence, for any , if is sufficiently large depending on and , then must contain an arithmetic progression of length . A byproduct of our analysis is a (slightly) improved bound for the constant .
Paper Structure (8 sections, 9 theorems, 65 equations)

This paper contains 8 sections, 9 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Previous structural results
  3. Bounds for the Littlewood conjecture
  4. McGehee-Pigno-Smith test functions
  5. A lower bound for $\| \widehat{1_A}\|_1$
  6. Improved constants for Littlewood's conjecture
  7. A structural inverse result
  8. Speculations about stronger structure

Key Result

Theorem 1.2

Let $N$ be a sufficiently large positive integer. Let $\delta \in (0, \frac{1}{2}]$ and $K> 0$. If $A$ is a set of $N$ integers such that $\| \widehat{1_{A}}\|_1\leqslant K\log N$ then there is a subset $A'\subseteq A$ of size $\left\lvert A'\right\rvert\gg N^{1-\delta}$ such that $\omega[A']\gg (\d

Theorems & Definitions (21)

  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Corollary 1.4
  • proof
  • Remark
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 11 more