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 $\alpha \in (0,1)$.

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