Israel Journal of Mathematics

, Volume 207, Issue 1, pp 203–228 | Cite as

A multi-dimensional Szemerédi theorem for the primes via a correspondence principle

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Abstract

We establish a version of the Furstenberg-Katznelson multi-dimensional Szemerédi theorem in the primes P:= {2, 3, 5, …}, which roughly speaking asserts that any dense subset of P d contains finite constellations of any given rational shape. Our arguments are based on a weighted version of the Furstenberg correspondence principle, relative to a weight which obeys an infinite number of pseudorandomness (or “linear forms”) conditions, combined with the main results of a series of papers by Green and the authors which establish such an infinite number of pseudorandomness conditions for a weight associated with the primes. The same result, by a rather different method, has been simultaneously established by Cook, Magyar and Titichetrakun and more recently by Fox and Zhao.

Keywords

Natural Number Arithmetic Progression Correspondence Principle Transference Principle Banach Theorem 
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Copyright information

© Hebrew University of Jerusalem 2015

Authors and Affiliations

  1. 1.UCLA Department of MathematicsLos AngelesUSA
  2. 2.Department of MathematicsTechnion—Israel Institute of TechnologyHaifaIsrael

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