Strong sweeping out for block sequences and related ergodic averages (Q1305191)

We introduce a condition we call the \(B(\alpha)\) condition, and prove that if a sequence of measures satisfies the \(B(\alpha)\) condition then the measures have the strong sweeping out property. Let \(\{\nu_n\}\) be a sequence of probability measures and let \(0<\alpha<{1\over 4}\). The sequence of measures is said to satisfy the \(B(\alpha)\) condition if given \(\varepsilon\), \(0<\varepsilon<1\), and a positive integer \(L\), we can find \(k_1,k_2,\dots,k_L\) and sets \(J_1,J_2,\dots,J_L\) such that \(\nu_{k_j}(J_j)\geq 1-\varepsilon\) for \(j=1,2,\dots,L\), and such that for any sequence \(\{x_j\}^L_{j=1}\) of real numbers, we can find a real number \(\theta\) such that \(w\theta\in x_j+(\alpha,1-\alpha)+\mathbb{Z}\) for all \(w\in J_j\), \(j=1,2,\dots,L\). The condition is a generalization of the \(C(\alpha)\) condition that was introduced in the earlier paper by \textit{M. Akcoglu}, \textit{A. Bellow}, \textit{R. L. Jones}, \textit{V. Losert}, \textit{K. Reinhold-Larsson} and \textit{M. Wierdl} [Ergodic Theory Dyn. Syst. 16, No. 2, 207-253 (1996; Zbl 0851.47004)]. As an application of showing a sequence of measures satisfies the \(B(\alpha)\) condition, we consider the block sequences introduced by \textit{A. Bellow} and \textit{V. Losert} [Contemp. Math. 26, 49-60 (1984; Zbl 0587.28013)] and show that in several cases, not only does divergence occur, but that we also have strong sweeping out. Related averages are also considered.