Abstract 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 c N for some absolute constant c 0. We explore what structure A must have if 1A ₁ K N for some constant K. Under such an assumption, we prove, for instance, that A contains a subset A^ A with A^ N^0. 99 such that |A^ + A^ | K^O (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.
Bloom et al. (Sat,) studied this question.