Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. (English) Zbl 0870.11015

This paper contains some extremely general results about the existence of certain configurations in sets of positive density. The power of these is best illustrated by the following special case: given an integer-valued polynomial \(p\) subject to the condition \(p(0)=0\), any set of integers of positive upper density contains arbitrarily long arithmetical progressions whose differences are of the form \(p(n)\). Here \(p(n)=n\) gives Szemerédi’s celebrated theorem, while the existence of a pair of integers with difference \(p(n)\) was first proved in 1978 by H. Furstenberg and Y. Katznelson [J. Anal. Math. 34, 275–291 (1978; Zbl 0426.28014)].
The general result sounds as follows. Let \(S\subset\mathbb Z^l\) be a set of positive upper Banach density, and let \(p_{ij}\), \(1\leq i\leq k, 1\leq j\leq t\) be integer-valued polynomials satisfying \(p_{ij}(0)=0\). Then for any \(v_1, ..., v_t\in\mathbb Z^l\) there is an integer \(n\) and a \(u\in\mathbb Z^l\) such that \[ u + \sum _{j=1}^t p_{ij}(n) v_j \in S \] for each \(1\leq i\leq k\).
The proof proceeds via a reformulation as a polynomial ergodicity theorem for a collection of commuting measure preserving transformations on a probability space. An important tool of the proof, and an interesting result in itself, is the following ‘polynomial topological van der Waerden theorem’. Let \((X, \varrho )\) be a compact metric space, \(T_1, \ldots, T_t\) commuting homeomorphisms of \(X\) and \(p_{ij}\), \(1\leq i\leq k, 1\leq j\leq t\), integer-valued polynomials with \(p_{ij}(0)=0\). Then for every \(\varepsilon >0\) we can find an \(n\in \mathbb N\) and an \(x\in X\) such that \[ \varrho (T_1^{p_{i1}(n)}\cdots T_t^{p_{ir}(n)} x, x ) < \varepsilon \] for all \(i\leq k\).


11B30 Arithmetic combinatorics; higher degree uniformity
11B05 Density, gaps, topology
11B25 Arithmetic progressions
11C08 Polynomials in number theory
28D05 Measure-preserving transformations
05D10 Ramsey theory


Zbl 0426.28014
Full Text: DOI


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