Generalized Polynomials and Mild Mixing

McCutcheon, Randall ; Quas, Anthony

Canadian journal of mathematics, Tome 61 (2009) no. 3, pp. 656-673

Résumé

An unsettled conjecture of V. Bergelson and I. Håland proposes that if $(X,\,\mathcal{A},\,\mu ,\,T)$ is an invertible weak mixing measure preserving system, where $\mu (X)<\infty $ , and if ${{p}_{1}},{{p}_{2}},...,{{p}_{k}}$ are generalized polynomials (functions built out of regular polynomials via iterated use of the greatest integer or floor function) having the property that no ${{p}_{i}}$ , nor any ${{p}_{i}}-{{p}_{j,}}i\ne j$ , is constant on a set of positive density, then for any measurable sets ${{A}_{0}},{{A}_{1}},...,{{A}_{K}}$ , there exists a zero-density set $E\subset Z$ such that 1 $$\underset{n\notin E}{\mathop{\underset{n\to \infty }{\mathop{\lim }}\,}}\,\mu ({{A}_{0}}\cap {{T}^{p1(n)}}{{A}_{1}}\cap \ldots \cap {{T}^{pk(n)}}{{A}_{k}})=\prod\limits_{i=0}^{k}{\mu ({{A}_{i}}).}$$ We formulate and prove a faithful version of this conjecture for mildly mixing systems and partially characterize, in the degree two case, the set of families $\left\{ {{p}_{1}},{{p}_{2}},\,.\,.\,.\,,\,{{p}_{k}} \right\}$ satisfying the hypotheses of this theorem.

Détail

DOI : 10.4153/CJM-2009-035-3

Mots-clés : 37A25, 28D05

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