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Generalized Polynomials and Mild Mixing

Published online by Cambridge University Press:  20 November 2018

Randall McCutcheon  and
Anthony Quas
Randall McCutcheon
Affiliation:
Department of Mathematical Sciences, University ofMemphis, Memphis, TN, USA, [email protected]
Anthony Quas
Affiliation:
Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada, [email protected]
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Abstract

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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.

Keywords

37A2528D05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 3 , 01 June 2009 , pp. 656 - 673
Copyright
Copyright © Canadian Mathematical Society 2009

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