Geometric and Functional Analysis

, Volume 25, Issue 3, pp 733–762 | Cite as

A relative Szemerédi theorem

  • David ConlonEmail author
  • Jacob Fox
  • Yufei Zhao


The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemerédi theorem which says that any subset of a pseudorandom set of integers of positive relative density contains long arithmetic progressions. In this paper, we give a simple proof of a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemerédi theorem for k-term arithmetic progressions in pseudorandom subsets of \({\mathbb{Z}_N}\) of density \({N^{-c_k}}\). The key component in our proof is an extension of the regularity method to sparse pseudorandom hypergraphs, which we believe to be interesting in its own right. From this we derive a relative extension of the hypergraph removal lemma. This is a strengthening of an earlier theorem used by Tao in his proof that the Gaussian primes contain arbitrarily shaped constellations and, by standard arguments, allows us to deduce the relative Szemerédi theorem.


Arithmetic Progression Sparse Graph Discrepancy Pair Regularity Lemma Removal Lemma 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Mathematical InstituteOxfordUK
  2. 2.Department of MathematicsMITCambridgeUSA

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