A shrinking target theorem for ergodic transformations of the unit interval

Abstract

We show that for any ergodic Lebesgue measure preserving transformation $f:[0,1)\to [0,1)$ and any decreasing sequence $\{b_i{\}}_{i=1}^{\infty }$ of positive real numbers with divergent sum, the set $$\underset{n=1}{\stackrel{\infty }{\cap }}\underset{i=n}{\stackrel{\infty }{\cup }}{f}^{-i}(B(R_{\alpha }^{i}x,b_i))$$ 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 $R_{\alpha }:[0,1)\to [0,1)$ 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.

arXiv commentJournal version. To appear in Discrete and Continuous Dynamical Systems