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A quantitative ergodic theory proof of Szemerédi’s theorem. (English) Zbl 1127.11011
The paper starts with a short discussion of some main ideas on which the known proofs of Szemerédi’s theorem on arithmetic progressions of length $$k$$ are based (the combinatorial ones, proofs using ergodic theory, Gowers’ proof using Fourier theory and the proof based on the hypergraph regularity lemma). This can be not only useful for an interested beginner in the subject, but also explains the author’s motivation for a new ergodic proof of Szemerédi’s $$k$$-term arithmetic subprogressions theorem given in the paper. The proof does not depend on the axiom of choice, nor on an infinite set of measures, the use of Fourier transform or inverse theorems of additive combinatorics. The author characterizes the proof as a “finitary” and “quantitative” version of the ergodic Furstenberg proof. The reading of the paper does not require familiarity with other proofs, but the author discusses on many places their relationship to the presented one. The proof also implies explicit quantitative bounds, though very poor.

##### MSC:
 11B25 Arithmetic progressions 05C75 Structural characterization of families of graphs 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 37B20 Notions of recurrence and recurrent behavior in dynamical systems
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