On the structure of sets with few three-term arithmetic progressions (Q1960282)

Summary: Fix a prime \(p\geq3\), and a real number \(0 <\alpha\leq1\). Let \(S\subset\mathbb F^n_p\) be any set with the least number of solutions to \(x + y = 2z\) (note that this means that \(x, z, y\) is an arithmetic progression), subject to the constraint that \(|S| \geq \alpha p^n\). What can one say about the structure of such sets \(S\)? In this paper, we show that they are ``essentially'' the union of a small number of cosets of some large-dimensional subspace of \(\mathbb F^n_p\).