A shrinking target theorem for ergodic transformations of the unit interval

Abstract

<p style='text-indent:20px;'>We show that for <i>any</i> ergodic Lebesgue measure preserving transformation <inline-formula><tex-math id="M1">\begin{document}$ f: [0,1) \rightarrow [0,1) $\end{document}</tex-math></inline-formula> and <i>any</i> decreasing sequence <inline-formula><tex-math id="M2">\begin{document}$ \{b_i\}_{i=1}^{\infty} $\end{document}</tex-math></inline-formula> of positive real numbers with divergent sum, the set</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ {\underset{n=1}{\overset{\infty}{\cap}} \, {\underset{i=n}{\overset{\infty}{\cup}}}\,} f^{-i}(B (R_{\alpha}^{i} x,b_i)) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>has full Lebesgue measure for almost every <inline-formula><tex-math id="M3">\begin{document}$ x \in [0,1) $\end{document}</tex-math></inline-formula> and almost every <inline-formula><tex-math id="M4">\begin{document}$ \alpha \in [0,1) $\end{document}</tex-math></inline-formula>. Here <inline-formula><tex-math id="M5">\begin{document}$ B(x,r) $\end{document}</tex-math></inline-formula> is the ball of radius <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula> centered at <inline-formula><tex-math id="M7">\begin{document}$ x \in [0,1) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ R_{\alpha}: [0,1) \rightarrow [0,1) $\end{document}</tex-math></inline-formula> is rotation by <inline-formula><tex-math id="M9">\begin{document}$ \alpha \in [0,1) $\end{document}</tex-math></inline-formula>. As a corollary, we provide partial answer to a question asked by Chaika (Question <inline-formula><tex-math id="M10">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>, [<xref ref-type="bibr" rid="b2">2</xref>]) in the context of interval exchange transformations.</p>