Remarks on the inverse Littlewood conjecture

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 $|A|=N$ then $\Vert \widehat{1_A}{\Vert }_1\ge c\log N$ for some absolute constant $c>0$. We explore what structure $A$ must have if $\Vert \widehat{1_A}{\Vert }_1\le K\log N$ for some constant $K$. Under such an assumption we prove, for instance, that $A$ contains a subset $A^{\prime }\subseteq A$ with $|A|\ge N^{0.99}$ such that $|A^{\prime }+A^{\prime }|\ll K^{O(1)}|A^{\prime }|$. As a consequence, for any $k\ge 3$, 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 $c$.

12 pages, to appear in Quart. J. Math. Thanks to a comment by the referee the value of c has increased to 0.17