Szemerédi's theorem with random differences and arithmetic Kakeya

Abstract

We prove that there exists an absolute constant $c > 0$ such that the following holds for every $k \ge 3$ and $p \ge n^{-c / \log \log k}$. Letting $δ> 0$ and $D \sim [n]_p$, with high probability every ${S \subset [n]}$ with $|S| \ge δn$ contains a $k$-term arithmetic progression whose common difference lies in $D$. This improves the best known bounds for the threshold of Szemerédi's theorem with random differences for all sufficiently large $k$, previously of the form ${p \ge n^{-2/k+o(1)}}$, due to Briët and Castro-Silva. We obtain our improvement via a small cover lemma, in the sense of Alon and Pham's work on random Cayley graphs, and its proof crucially relies on a construction of Green and Ruzsa in the context of the arithmetic Kakeya conjecture. The usage of the latter was suggested to us by GPT 6 Astra.