Approximate arithmetic structure in large sets of integers

Funding: JMF acknowledges financial support from an EPSRC Standard Grant (EP/R015104/1) and a Leverhulme Trust Research Project Grant (RPG-2019-034). HY was financially supported by the University of St Andrews and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.803711).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 Δ of the progression, we improve a previous result of o(Δ) to O(Δα) for any α∈(0,1).PostprintPeer reviewe