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szemerédi's theorem

Proceedings of the Edinburgh Mathematical Society (1)

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Szemerédi's Theorem in the Primes
Part of
- Sequences and sets
Luka Rimanić, Julia Wolf
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 62 / Issue 2 / May 2019

Published online by Cambridge University Press:

19 November 2018, pp. 443-457
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Green and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that, in fact, any subset of the primes of relative density tending to zero sufficiently slowly contains a three-term progression. This was followed by work of Helfgott and de Roton, and Naslund, who improved the bounds on the relative density in the case of three-term progressions. The aim of this note is to present an analogous result for longer progressions by combining a quantified version of the relative Szemerédi theorem given by Conlon, Fox and Zhao with Henriot's estimates of the enveloping sieve weights.