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Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory

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Abstract

We introduce two notions of complexity of a system of polynomials p 1,..., p r ∈ ℤ[n] and apply them to characterize the limits of the expressions of the form \(\mu (A_0 \cap T^{ - p_1 (n)} A_1 \cap \cdots \cap T^{ - p_r (n)} A_r )\) where T is a skew-product transformation of a torus \(\mathbb{T}^d \) and \(A_i \subseteq \mathbb{T}^d \) are measurable sets. The dynamical results obtained allow us to construct subsets of integers with specific combinatorial properties related to the polynomial Szemerédi theorem.

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, OH, 43210, USA

    V. Bergelson & A. Leibman

  2. Laboratoire de Mathématiques et Physique Théorique, UMR CNRS 6083, Université François Rabelais Tours, 37200, Tours, France

    E. Lesigne

Authors
  1. V. Bergelson
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  2. A. Leibman
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  3. E. Lesigne
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Corresponding author

Correspondence to V. Bergelson.

Additional information

Bergelson and Leibman were supported by NSF grants DMS-0345350 and DMS-0600042.

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Bergelson, V., Leibman, A. & Lesigne, E. Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory. J Anal Math 103, 47–92 (2007). https://doi.org/10.1007/s11854-008-0002-z

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  • DOI: https://doi.org/10.1007/s11854-008-0002-z

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