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.

• (i) Either there is a Jset H of O(log^c |A|) subgroups of G with |AΔ∪H| = o(|A|) where ∪H = ∪_H∈H H, or there is a character γ ∈ $\gamma \in \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over G}$ such that $- {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over 1} }_A}_A(\gamma ) = \Omega ({\log ^c}\left| A \right|)$ 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 |AΔH| = o(|A|), or there is a character γ ∈ $\gamma \in \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over G}$ such that $- {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over 1} }_A}_A(\gamma ) = \Omega ({\left| A \right|^{\Omega (1)}})$.