Bounds for multiple recurrence rate and dimension (Q2328040)

For a probability measure-preserving system \((X,\mathcal{B},\mu,T)\) and a set \(A\in\mathcal{B}\) with \(\mu(A)>0\), the Poincaré recurrence theorem states that \(\mu\)-almost every point in \(A\) returns to \(A\) infinitely often. If the space \(X\) has a compatible metric \(d\) for which \(\mathcal{B}\) is the Borel \(\sigma\)-algebra then a result of \textit{M. D. Boshernitzan} [Invent. Math. 113, No. 3, 617--631 (1993; Zbl 0839.28008)] gives qualitative information about the closeness of returns, showing that \(\liminf_{n\to\infty}d(x,T^nx)=0\) for \(\mu\)-almost every \(x\in X\) and, under the geometric hypothesis that the \(\alpha\)-Hausdorff measure of \(X\) is finite for some \(\alpha>0\), goes on to show that \(\liminf_{n\to\infty}\bigl(n^{1/\alpha}d(x,T^nx)\bigr)<\infty\) for \(\mu\)-almost every \(x\in X\). \textit{L. Barreira} and \textit{B. Saussol} [Commun. Math. Phys. 219, No. 2, 443--463 (2001; Zbl 1007.37012)] gave estimates for the first return time to a metric ball, and \textit{D. H. Kim} [Nonlinearity 22, No. 1, 1--9 (2009; Zbl 1167.37006)] generalized Boshernitzan's results to actions of countable discrete groups. Here a multiple and simultaneous analogue of these results are found; the results are too complicated to state here but the flavor is to find quantitative versions of statements of the form \(\liminf_{n\to\infty}\mathrm{diam} \{x,T^nx,T^{2n}x,\dots,T^{Ln}x\}=0\) for \(\mu\)-almost every \(x\in X\). The methods used are diverse and they include results of \textit{W. T. Gowers} [Geom. Funct. Anal. 11, No. 3, 465--588 (2001); Erratum 11, No. 4, 869 (2001; Zbl 1028.11005)] on the quantitative Szemerédi theorem to study long simultaneous return.