Dynamical covering problems on the triadic Cantor set

作者：Wang, Bao-Wei^*; Wu, Jun; Xu, Jian

来源：Comptes Rendus Mathematique, 2017, 355(7): 738-743.

DOI：10.1016/j.crma.2017.05.014

• DIOPHANTINE APPROXIMATION
• CIRCLE
• ARCS

In this note, we consider the metric theory of the dynamical covering problems on the triadic Cantor set K. More precisely, let Tx = 3x (mod 1) be the natural map on K, mu the standard Cantor measure and x(0) is an element of K a given point. We consider the size of the set of points in K which can be well approximated by the orbit {T(n)x(0)}(n >= 1) of x(0), namely the set @@@ D(x(0), phi) := {y is an element of K : |T(n)x(0) - y| < phi(n) for infinitely many n is an element of N}, @@@ where phi is a positive function defined on N. It is shown that for mu almost all x(0) is an element of K, the Hausdorff measure of D(x(0), phi) 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.

• 出版日期2017-7
• 单位华中科技大学