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