Remarks on the inverse Littlewood conjecture

TL;DR

This paper investigates the structure of finite integer sets with small Fourier transform norms, showing such sets contain large structured subsets and contain long arithmetic progressions, refining bounds related to the inverse Littlewood conjecture.

Contribution

It demonstrates that sets with small Fourier norms must contain large structured subsets and long arithmetic progressions, providing new bounds and insights into the inverse Littlewood conjecture.

Findings

Sets with small Fourier norms contain large structured subsets.

Such sets necessarily include long arithmetic progressions.

The bounds on the constant in the Littlewood conjecture are slightly improved.

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 $c$.