Almost arithmetic progressions in the primes and other large sets

Jonathan M. Fraser

doi:10.1080/00029890.2019.1586264

Almost arithmetic progressions in the primes and other large sets

Jonathan M. Fraser

Pure Mathematics

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

Abstract

A celebrated and deep result of Green and Tao states that the primes contain arbitrarily long arithmetic progressions. In this note, I provide a straightforward argument demonstrating that the primes get arbitrarily close to arbitrarily long arithmetic progressions. The argument also applies to “large sets” in the sense of the Erdős conjecture on arithmetic progressions. The proof is short, completely self-contained, and aims to give a heuristic explanation of why the primes, and other large sets, possess arithmetic structure.

Original language	English
Pages (from-to)	553-558
Number of pages	6
Journal	The American Mathematical Monthly
Volume	126
Issue number	6
Early online date	29 May 2019
DOIs	https://doi.org/10.1080/00029890.2019.1586264
Publication status	Published - May 2019

Keywords

Arithmetic progression
Primes
Green-Tao Theorem
Erdős-Turan Conjecture

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10.1080/00029890.2019.1586264

almostAPs
Copyright © 2019 The Mathematical Association of America. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version 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/10.1080/00029890.2019.1586264
Accepted author manuscript, 210 KB

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
EPSRC
1/02/18 → 11/06/21
Project: Standard
Fractal Geometry and Dimension: Fractal Geometry and dimension theory
Fraser, J. (PI)
The Leverhulme Trust
1/09/16 → 30/06/18
Project: Fellowship

Cite this

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Almost arithmetic progressions in the primes and other large sets

Abstract

Keywords

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Projects

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis

Fractal Geometry and Dimension: Fractal Geometry and dimension theory

Cite this