A shrinking target theorem for ergodic transformations of the unit interval

TL;DR

This paper proves a shrinking target theorem for ergodic transformations on the unit interval, showing that certain sets related to shrinking neighborhoods have full measure for almost every point and rotation, with implications for interval exchange transformations.

Contribution

It establishes a new shrinking target result for ergodic Lebesgue measure-preserving transformations, extending previous work and addressing a question by Chaika on interval exchange transformations.

Findings

Sets related to shrinking targets have full measure for almost every point and rotation.

The result applies to ergodic Lebesgue measure-preserving transformations.

Provides partial answers to open questions in the theory of interval exchange transformations.

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.