Ergodic theory and diophantine problems (Q2711285)

The author investigates ergodic theory approach to Ramsey theory and to some diophantine problems. This interesting survey includes some recent results on a remarkable generalization of Szemerédi's theorem due to the author under review and \textit{A. Leibman} [J. Am. Math. Soc. 9, No. 3, 725-753 (1996; Zbl 0870.11015)]. It is a continuation of the article by \textit{H. Furstenberg} [J. Anal. Math. 31, 204-256 (1977; Zbl 0347.28016)] and some other articles of H. Furstenberg and his co-authors. Another approach to some diophantine problems which is based on ergodic methods and elements of Fourier analysis is described in the monograph of \textit{A. Postnikov} [Proc. Steklov Inst. Math. 82 (1966; Zbl 0178.04702)]. A very interesting quantitative approach to Szemerédi's theorem and to Ramsey theory is proposed by \textit{W. T. Gowers} [Proc. ICM, Berlin, Doc. Math., Extra volume ICM 1998, Vol. I, 617-629 (1998)]. NEWLINENEWLINENEWLINEAfter some introductory section, second section of the paper under review considers some diophantine problems related to polynomials and their connections with combinatorics and dynamics. Third section concerns partition (geometric) Ramsey theory and topological dynamics. Fourth section concerns density Ramsey theory and ergodic theory of multiple recurrence. Section five gives polynomial ergodic theorems and their connection with Ramsey theory. Basic ideas and techniques that presented in the sections are treated in detail, including exercises. The appendix contains ultrafilter and elementary proofs of a special case of Weyl's theorem on the equidistribution of polynomials.NEWLINENEWLINEFor the entire collection see [Zbl 0942.00028].