2021 Approximate arithmetic structure in large sets of integers
Jonathan M. Fraser, Han Yu
Author Affiliations +
Real Anal. Exchange 46(1): 163-174 (2021). DOI: 10.14321/realanalexch.46.1.0163

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 $\alpha \in (0,1)$. This improvement comes from a new approach relying on an iterative application of Szemerédi's Theorem.

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Jonathan M. Fraser. Han Yu. "Approximate arithmetic structure in large sets of integers." Real Anal. Exchange 46 (1) 163 - 174, 2021. https://doi.org/10.14321/realanalexch.46.1.0163

Information

Published: 2021
First available in Project Euclid: 14 October 2021

Digital Object Identifier: 10.14321/realanalexch.46.1.0163

Subjects:
Primary: 11B25
Secondary: 11B05

Keywords: arithmetic progressions , Erdős conjecture

Rights: Copyright © 2021 Michigan State University Press

Vol.46 • No. 1 • 2021
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