Remarks on the Inverse Littlewood Conjecture

The Littlewood conjecture, proven by Konyagin and McGehee–Pigno–Smith in the 1980s, states that if A⊂ Z is a finite set of integers with A =N, then 1A ₁ clog N for some absolute constant c > 0. We explore what structure A must have if 1A ₁ Klog N for some constant K. Under such an assumption, we prove, for instance, that A contains a subset A⊆ A with A N0.99 such that |A + A| KO(1)|A|. As a consequence, for any k 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.