A relative Szemer,di theorem

作者：Conlon David^*; Fox Jacob; Zhao Yufei

来源：Geometric and Functional Analysis, 2015, 25(3): 733-762.

DOI：10.1007/s00039-015-0324-9

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 Szemer,di 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 Szemer,di theorem, showing that a much weaker pseudorandomness condition is sufficient. Our strengthened version can be applied to give the first relative Szemer,di theorem for k-term arithmetic progressions in pseudorandom subsets of of density . 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 Szemer,di theorem.

• 出版日期2015-6
• 单位MIT