A dichotomy law for the Diophantine properties in ${\hat{I}}^2$-dynamical systems

Abstract

Let ${\hat{I}}^2>1$ be a real number and define the ${\hat{I}}^2$-transformation on $[0,1]$ by $T_{\hat{I}}^2:x\mapsto {\hat{I}}^{2}x\mathrm{mod}1$. Further, define $$W_y(T_β,Ψ):=\{x\in [0, 1]:|T_β^nx-y|<Ψ(n) \mbox{ for infinitely many $n$}\}$$ and $$W(T_β,Ψ):=\{(x, y)\in [0, 1]^2:|T_β^nx-y|<Ψ(n) \mbox{ for infinitely many $n$}\},$$ where $\hat{I}¨:\mathbb{N}\to {\mathbb{R}}_{>0}$ is a positive function such that $\hat{I}¨(n)\to 0$ as $n\to \infty$. In this paper, we show that each of the above sets obeys a Jarník-type dichotomy, that is, the generalised Hausdorff measure is either zero or full depending upon the convergence or divergence of a certain series. This work completes the metrical theory of these sets.

Accepted for publication in Mathematika