Remarks on the inverse Littlewood conjecture

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 $c$.