Remarks on the Inverse Littlewood Conjecture

Journal article

The Littlewood conjecture, proven by Konyagin and McGehee–Pigno–Smith in the 1980s, states that if is a finite set of integers with , then for some absolute constant . We explore what structure A must have if for some constant K. Under such an assumption, we prove, for instance, that A contains a subset with such that . As a consequence, for any , 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.

Authors: Bloom, TF; Green, B

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Oxford University Press
• Journal: The Quarterly Journal of Mathematics
• Article number: haag016
• Publication date: 2026-06-01
• Acceptance date: 2026-05-08
• DOI: 10.1093/qmath/haag016
• EISSN: 1464-3847
• ISSN: 0033-5606
• Language: English