117 research outputs found
Set-polynomials and polynomial extension of the Hales-Jewett Theorem
An abstract, Hales-Jewett type extension of the polynomial van der Waerden
Theorem [J. Amer. Math. Soc. 9 (1996),725-753] is established:
Theorem. Let r,d,q \in \N. There exists N \in \N such that for any r-coloring
of the set of subsets of V={1,...,N}^{d} x {1,...,q} there exist a set a
\subset V and a nonempty set \gamma \subseteq {1,...,N} such that a \cap
(\gamma^{d} x {1,...,q}) = \emptyset, and the subsets a, a \cup (\gamma^{d} x
{1}), a \cup (\gamma^{d} x {2}), ..., a \cup (\gamma^{d} x {q}) are all of the
same color.
This ``polynomial'' Hales-Jewett theorem contains refinements of many
combinatorial facts as special cases. The proof is achieved by introducing and
developing the apparatus of set-polynomials (polynomials whose coefficients are
finite sets) and applying the methods of topological dynamics.Comment: 43 pages, published versio
Van der Corput sets in Z^d
In this partly expository paper we study van der Corput sets in , with
a focus on connections with harmonic analysis and recurrence properties of
measure preserving dynamical systems. We prove multidimensional versions of
some classical results obtained for in \cite{K-MF} and \cite{R},
establish new characterizations, introduce and discuss some modifications of
van der Corput sets which correspond to various notions of recurrence, provide
numerous examples and formulate some natural open questions
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