Abstract view
Generalized Polynomials and Mild Mixing


Published:20090601
Printed: Jun 2009
Randall McCutcheon
Anthony Quas
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Abstract
An unsettled conjecture of V. Bergelson and I. H\aa land proposes that
if $(X,\alg,\mu,T)$ is an invertible weak mixing measure preserving
system, where $\mu(X)<\infty$, and if $p_1,p_2,\dots ,p_k$ 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 $p_ip_j$, $i\neq j$, is constant on a
set of positive density, then for any measurable sets
$A_0,A_1,\dots
,A_k$, there exists a zerodensity set $E\subset \z$ such that
\[\lim_{\substack{n\to\infty\\ n\not\in E}} \,\mu(A_0\cap T^{p_1(n)}A_1\cap \cdots
\cap T^{p_k(n)}A_k)=\prod_{i=0}^k \mu(A_i).\] 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
$\{ p_1,p_2, \dots ,p_k\}$ satisfying the hypotheses of this theorem.