Almost arithmetic progressions in the primes and other large sets
Classical Analysis and ODEs
2019-09-20 v2 Combinatorics
History and Overview
Metric Geometry
Number Theory
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 Erd\H{o}s-Tur\'an. 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.
Cite
@article{arxiv.1809.01409,
title = {Almost arithmetic progressions in the primes and other large sets},
author = {Jonathan M. Fraser},
journal= {arXiv preprint arXiv:1809.01409},
year = {2019}
}
Comments
Expository article, 6 pages, 1 figure. To appear in The American Mathematical Monthly