The classic Poincare recurrence theorem states that for certain measure-preserving dynamical systems, generic points, after sufficiently long but finite iterations of T, will return to a neighborhood arbitrarily close to themselves. This is a qualitative result with no quantitative information. In 1993, Boshernitzan obtained a quantitative result which relates the recurrence rate to the Hausdorff dimension of the metric space. This result was further refined in some dynamical systems, especially conformal dynamical systems with nice mixing properties. In this talk, we will discuss the quantitative recurrence properties for piecewise expanding maps (non-conformal dynamical systems). Part of this work is motivated by the classical theories of weighted Diophantine approximation and multiplicative Diophantine approximation. This is a joint work with Prof. Lingmin Liao.