An arithmetic transference proof of a relative Szemerédi theorem. (March 2014)

• Record Type: Journal Article
• Authors: ZHAO, YUFEI
• Is Part Of: Mathematical proceedings of the Cambridge Philosophical Society. Volume 156:Part 2(2014:Mar.)
• Page Start: 255
• Page End: 261
• Publication Date: 2014-03
• Subjects: Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0305004113000662
• Languages: English
• ISSNs: 0305-0041

Abstract

<abstract abstract-type="normal"> <title>Abstract</title> <p>Recently, Conlon, Fox and the author gave a new proof of a relative Szemerédi theorem, which was the main novel ingredient in the proof of the celebrated Green–Tao theorem that the primes contain arbitrarily long arithmetic progressions. Roughly speaking, a relative Szemerédi theorem says that if <italic>S</italic> is a set of integers satisfying certain conditions, and <italic>A</italic> is a subset of <italic>S</italic> with positive relative density, then <italic>A</italic> contains long arithmetic progressions, and our recent results show that <italic>S</italic> only needs to satisfy a so-called linear forms condition.</p> <p>This paper contains an alternative proof of the new relative Szemerédi theorem, where we directly transfer Szemerédi's theorem, instead of going through the hypergraph removal lemma. This approach provides a somewhat more direct route to establishing the result, and it gives better quantitative bounds.</p> <p>The proof has three main ingredients: (1) a transference principle/dense model theorem of Green–Tao and Tao–Ziegler (with simplified proofs given later by Gowers, and independently, Reingold–Trevisan–Tulsiani–Vadhan) applied with a discrepancy/cut-type norm (instead of a Gowers uniformity norm as it was applied in earlier works); (2) a counting lemma established by Conlon, Fox and the author; and (3) Szemerédi's theorem as a black box.</p> </abstract>