Note on weakly mixing transformations (Q1188296)

Let \((X,d)\) be a metric space with a probability measure \(P\) defined on a \(\sigma\)-algebra \(F\) of subsets of \(X\) including the Borel sets and \(T: X\to X\) be measure preserving. The diameter of a set \(A\subset X\) is defined by \(\text{diam}(A)=\sup\{d(x,y): x,y\in A\}\). The essential diameter of a measurable \(A\) is defined by \(\text{ess diam}(A)=\inf\{\text{diam}(B): B\in F\) and \(P(B)=P(A)\}\). The geometric diameter of order \(k\) \((k\geq 2)\) is defined by \[ D_ k(A)=\sup\left\{\prod_{1\leq i<j\leq k} d(x_ i,x_ j)^{2/k(k-1)}: x_ 1,\dots,x_ k\in A\right\}. \] Note that \(D_ 2(A)=\text{diam}(A)\). The author essentially extends theorems of \textit{R. E. Rice} [Aequationes Math. 17, 104-108 (1978; Zbl 0398.28020)] and \textit{C. Sempi} [Rad. Mat. 1, No. 1, 3-7 (1985; Zbl 0587.28015)]. Theorem 1 (Rice): In the above situation if every open set has positive measure and if \(T\) is strongly mixing and \(A\) is measurable with \(P(A)>0\), then (i) \(\lim_{n\to\infty}\text{diam}(T^ n A)=\text{diam}(X)\), (ii) If in addition \(TB\in F\) whenever \(B\in F\), then \(\lim_{n\to\infty}\text{ess diam}(T^ n A)=\text{diam}(X)\). Theorem 2 (Sempi): Under the same conditions as (ii) above but with \(T\) only weakly mixing, \(\limsup_{n\to\infty}\text{diam}(T^ n A)=\text{diam}(X)\). The present author shows that in Theorem 2 (a) the requirement that \(TB\in F\) whenever \(B\in F\) can be dropped and diam can be replaced by \(D_ k\) \((k\geq 2)\), and (b) diam can be replaced by ess diam.