Approximate arithmetic structure in large sets of integers

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