Let $K$ be a homogeneous self-similar set satisfying the strong separation condition. This paper is concerned with the quantitative recurrence properties of the natural map $T: K\rightarrow K$ induced by the shift. Let $\mu$ be the natural self-similar measure supported on $K$. For a positive function $\varphi$ defined on $\mathbb{ N}$, we show that the $\mu$-measure of the following set \begin{ equation*} R(\varphi):=\{ x\in K: |T^n x-x|<\varphi(n) \; \text{ for infinitely many} \; n\in\mathbb{ N}\} \end{ equation*} is null or full according to convergence or divergence of a certain series. Moreover, a similar dichotomy law holds for the general Hausdorff measure, which completes the metric theory of this set.
Categories: math.DS math.NT