Chowla’s cosine problem

Abstract

Suppose that G is a discrete abelian group and A ⊂ G is a finite symmetric set. We show two main results.(where c > 0 is the same absolute constant.(ii) If G is finite and |A| = Ω(|G|) then either there is a subgroup H G such that |A H| = o(|A|), or there is a character γ ∈ G such that − 1 A (γ) = Ω(|A| Ω(1) ).