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Generalized Polynomials and Mild Mixing

Open Access article
 Printed: Jun 2009
  • Randall McCutcheon
  • Anthony Quas
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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 $p_i-p_j$, $i\neq j$, is constant on a set of positive density, then for any measurable sets $A_0,A_1,\dots ,A_k$, there exists a zero-density 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.
MSC Classifications: 37A25, 28D05 show english descriptions Ergodicity, mixing, rates of mixing
Measure-preserving transformations
37A25 - Ergodicity, mixing, rates of mixing
28D05 - Measure-preserving transformations

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