Approximate arithmetic structure in large sets of integers

Abstract

We prove that if a set is ‘large’ in the sense of Erdős, then it approximates arbitrarily long arithmetic progressions in a strong quantitative sense. More specifically, expressing the error in the approximation in terms of the gap length $\Delta$ of the progression, we improve a previous result of $o(\Delta)$ to $O(\Delta^\alpha)$ for any $\alpha \in (0,1)$. This improvement comes from a new approach relying on an iterative application of Szemerédi's Theorem.