We give an upper bound for the rate of simultaneous recurrence for some commuting family of measure preserving transformations. This provides a generalization of the Boshernitzan quantitative recurrence theorem for single transformation. We also give a lower bound for the Hausdorff dimension of the invariant measures for such transformations and for countable group actions.
Michihiro HIRAYAMA. "Bounds for Multiple Recurrence Rate and Dimension." Tokyo J. Math. 42 (1) 239 - 253, June 2019. https://doi.org/10.3836/tjm/1502179281