Approximate arithmetic structure in large sets of integers

Jonathan Fraser; Han Yu

doi:10.14321/realanalexch.46.1.0163

Approximate arithmetic structure in large sets of integers

Jonathan Fraser, Han Yu

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

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 Δ of the progression, we improve a previous result of o(Δ) to O(Δα) for any α∈(0,1).

Original language	English
Pages (from-to)	163-174
Journal	Real Analysis Exchange
Volume	46
Issue number	1
Early online date	14 Oct 2021
DOIs	https://doi.org/10.14321/realanalexch.46.1.0163
Publication status	Published - 2021

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10.14321/realanalexch.46.1.0163

Fraser-2020-Approximate-arithmetic-structure-REA-AAM
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.
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https://projecteuclid.org/journals/real-analysis-exchange/volume-46/issue-1/Approximate-arithmetic-structure-in-large-sets-of-integers/10.14321/realanalexch.46.1.0163.short

Cite this

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abstract = "We prove that if a set is `large' in the sense of Erd{\H o}s, 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). ",

author = "Jonathan Fraser and Han Yu",

note = "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{\textquoteright}s Horizon 2020 research and innovation programme (grant agreement No.803711).",

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N2 - 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).

AB - 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).

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Approximate arithmetic structure in large sets of integers

Abstract

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Approximate arithmetic structure in large sets of integers

Abstract

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New perspectives in the dimension: New perspectives in the dimension theory of fractals

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

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