Recurrence in mean (Q1810229)

Let \((X,d)\) be a metric space. Using a generalization \(H_h\) of the standard Hausdorff measure \(H_\alpha\) (coinciding with \(H_\alpha\) for \(h(t)=t^\alpha\)), the author shows that for a discrete time dynamical system \((T,X,\mu)\), such that \(H_h(X)<\infty\) and any \(\mu\)-measurable set is \(H_h\)-measurable, the \(\mu\)-average value of the recurrence constant \(C(x):=\liminf_{n\to\infty} nh(d(T^nx,x))\) over any measurable set \(A\) cannot exceed \(H_h(A)\). This estimate generalizes results by \textit{M. D. Boshernitzan} [Invent. Math. 113, 617-631 (1993; Zbl 0839.28008)] obtained in the case of \(H_\alpha\).