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 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).

Citation

Fraser , J & Yu , H 2021 , ' Approximate arithmetic structure in large sets of integers ' , Real Analysis Exchange , vol. 46 , no. 1 , pp. 163-174 . https://doi.org/10.14321/realanalexch.46.1.0163

Publication

Real Analysis Exchange

Status

Peer reviewed

DOI

https://doi.org/10.14321/realanalexch.46.1.0163

ISSN

0147-1937

Type

Journal article

Rights

Copyright © 2021 Real Exchange Analysis. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.14321/realanalexch.46.1.0163.

Description

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).