Arithmetic progressions -- an operatortheoretic view

Tanja Eisner; Rainer Nagel

doi:10.3934/dcdss.2013.6.657

Article Contents

2013, Volume 6, Issue 3: 657-667. Doi: 10.3934/dcdss.2013.6.657

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Arithmetic progressions -- an operator theoretic view

Tanja Eisner^1, and
Rainer Nagel^2,

1.
KdV Institute for Mathematics, University of Amsterdam, P.O. Box 94248, 1090 GE Amsterdam
2.
University of Tübingen, Mathematics Institute, Auf der Morgenstelle 10, D-72076 Tübingen

Received: February 2010

Revised: May 2010

Published: June 2013

Abstract / Introduction Related Papers Cited by

Abstract

Motivated by the recent Green--Tao theorem on arithmetic progressions in the primes, we discuss some of the basic operator theoretic techniques used in its proof. In particular, we obtain a complete proof of Szemerédi's theorem for arithmetic progressions of length $3$ (Roth's theorem) and the Furstenberg--Sárközy theorem.

Keywords:

Arithmetic progressions,
multiple ergodic theorems,
Roth's theorem,
operator theoretic methods,
Jacobs-Glicksberg-deLeeuw decomposition.

Mathematics Subject Classification: 47A35, 37A45.

Citation:

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