Note on weakly mixing transformations

Summary LetT be a weakly mixing transformation with respect to a probability measureP on a metric space (X, d). Suppose further that every open ball of (X, d) has positive measure. Then we show that, for anyP-measurable setA withP(A) > 0, lim supD_k(TⁿA) =D_k(X) fork = 2, 3,, whereD_k(B) is the geometric diameter of orderk of a subsetB ofX. It is shown further that D_k can be replaced by essD_k, in the case whenTB is measurable wheneverB is measurable. These results complement a previous one due to R. E. Rice for strongly mixing transformations and improve a result of C. Sempi on weakly mixing transformations.