A shrinking target theorem for ergodic transformations of the unit interval

Abstract

We show that for any ergodic Lebesgue measure preserving transformation $f: [0,1) \rightarrow [0,1)$ and any decreasing sequence $\{b_i\}_{i=1}^{\infty}$ of positive real numbers with divergent sum, the set $$\underset{n=1}{\overset{\infty}{\cap}} \, \underset{i=n}{\overset{\infty}{\cup}}\, f^{-i}(B (R_{\alpha}^{i} x,b_i))$$ has full Lebesgue measure for almost every $x \in [0,1)$ and almost every $\alpha \in [0,1)$. Here $B(x,r)$ is the ball of radius $r$ centered at $x \in [0,1)$ and $R_{\alpha}: [0,1) \rightarrow [0,1)$ is rotation by $\alpha \in [0,1)$. As a corollary, we provide partial answer to a question asked by Chaika in the context of interval exchange transformations.