Dynamical covering problems on the triadic Cantor set

Abstract

In this note, we consider the metric theory of the dynamical covering problems on the triadic Cantor set K. More precisely, let Tx=3x(mod1) be the natural map on K, μ the standard Cantor measure and x0∈K a given point. We consider the size of the set of points in K which can be well approximated by the orbit {Tnx0}n≥1 of x0, namely the setD(x0,φ):={y∈K:|Tnx0−y|<φ(n)for infinitely manyn∈N}, where φ is a positive function defined on N. It is shown that for μ almost all x0∈K, the Hausdorff measure of D(x0,φ) is either zero or full depending upon the convergence or divergence of a certain series. Among the proof, as a byproduct, we obtain an inhomogeneous counterpart of Levesley, Salp and Velani's work on a Mahler's question about the Diophantine approximation on the Cantor set K.