Chowla’s Cosine Problem

Suppose that G is a discrete abelian group and A ⊂ G is a finite symmetric set. We show two main results. (i) Either there is a set H of O(log |A|) subgroups of G with |A△ S H| = o(|A|), or there is a character γ ∈ b G such that −c 1A(γ) = Ω(log c |A|) where c > 0 is the same absolute constant. (ii) If G is finite and |A| = Ω(|G|) then either there is a subgroup H 6 G such that |A△H| = o(|A|), or there is a character γ ∈ b G such that −c 1A(γ) = Ω(|A| Ω(1)).