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Results of Wyner and Ziv and of Ornstein and Weiss show that if one observesthe first k outputs of a finite-valued ergodic process, then the waiting timeuntil this block appears again is almost surely asymptotic to $2^{hk}$, where$h$ is the entropy of the process. We examine this phenomenon when the allowedreturn times are restricted to some subset of times, and generalize the resultsto processes parameterized by other discrete amenable groups.  We also obtain a uniform density version of the waiting time results: For aprocess on $s$ symbols, within a given realization, the density of the initial$k$-block within larger $n$-blocks approaches $2^{-hk}$, uniformly in $n>s^k$,as $k$ tends to infinity. Again, similar results hold for processes with otherindexing groups.Comment: To appear in Journal d'Analyse Mathematiqu

Return times, recurrence densities and entropy for actions of some discrete amenable groups