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

Vitaly Bergelson (1) , Alexander Leibman (1) , Emmanuel Lesigne (2)

1 Department of mathematics, OSU
2 LMPT - Laboratoire de Mathématiques et Physique Théorique

We introduce two notions of complexity of a system of polynomials $p_{1},\ld,p_{r}\in\Z[n]$ and apply them to characterize the limits of the expressions of the form $\mu(A_{0}\cap T^{-p_{1}(n)}A_{1}\cap\ld\cap T^{-p_{r}(n)}A_{r})$ where $T$ is a skew-product transformation of a torus $\T^{d}$ and $A_{i}\sle\T^{d}$ are measurable sets. The obtained dynamical results allow us to construct subsets of integers with specific combinatorial properties related to the polynomial Szemer\'{e}di theorem.

Mots clés: recurrence; combinatorial number theory; polynomial Szemeredi theorem

Domaines: Systèmes dynamiques [math.DS]

Vitaly Bergelson, Alexander Leibman, Emmanuel Lesigne. Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory. Journal d'analyse mathématique, 2007, 103, pp.47-92. ⟨hal-00017730v2⟩