New bounds on cap sets (Q2879894)

A set \(A\subseteq \mathbb F^n_3\) is called a cap set if it contains no solutions of the equation \(x+ y+ 7= 0\), \(x,y,7\in A\), \(x\neq y\neq 7\). These sets correspond to arithmetic progressions of the length three in \(\mathbb Z\). Improving a result of R. Meshulom the authors prove that any cap set has size less than \(O({3^n\over n^{1+\varepsilon}})\), where \(\varepsilon> 0\) is an absolute (but small) constant. It corresponds to the famous conjecture of Erdős and Turán on sets without arithmetic progressions. The proof based on new structural results about sets with a nontrivial lower bounds for the additive energy as well as new statements on the spetrum of cap sets.