On Systems of Complexity One in the Primes

Henriot, Kevin

doi:10.1017/s001309151500053x

Cited by 1 publication

(6 citation statements)

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“…By Proposition 6.2 in [Hen16], since M is large enough and equal to N ′ up to a constant factor, there exists a pseudorandom majorant ν :…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…Note that [11], since M is large enough and equal to N up to a constant factor, there exists a pseudorandom majorant ν : (1) , as well as the inequality…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…The existence of such a pseudorandom majorant was established already in [GT08] using ingredients from [GPY09]. It is here that we make crucial use of the simplified and optimised estimates from [Hen16], allowing us to choose w as large as c 0 log N ′ .…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…Define the corresponding critical densities by Regarding relative density in the primes, the current record for progressions of length 3 is α 3 (P N ) ≪ (log log N ) −1+o(1) , arrived at through a series of articles by Green [Gre05], Helfgott and de Roton [HdR11], and finally Naslund [Nas15]. Henriot [Hen16] extended Naslund's result to all linear systems of complexity one, and our result relies crucially on the optimised estimates of the enveloping sieve weights he gave in this paper. The aim of this note is to extend these results to longer arithmetic progressions as follows.…”

Section: Introductionmentioning

confidence: 99%

“…The first part consists of a relative Szemerédi theorem as proved by Conlon, Fox and Zhao [CFZ14], which we quantify for our purposes in Section 2 (some details have been relegated to the appendix). In the second part we make use of Henriot's [Hen16] optimised estimates of the usual sieve weights associated with the primes.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Szemerédi's Theorem in the Primes

Rimanic¹,

Wolf²

2018

Proceedings of the Edinburgh Mathematical Society

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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 3-term 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 3-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.Date: September 15, 2017.

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“…By Proposition 6.2 in [Hen16], since M is large enough and equal to N ′ up to a constant factor, there exists a pseudorandom majorant ν :…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…Note that [11], since M is large enough and equal to N up to a constant factor, there exists a pseudorandom majorant ν : (1) , as well as the inequality…”

Section: Proof Of Theorem 11mentioning

confidence: 99%