On Dirichlet non-improvable numbers and shrinking target problems

Abstract

In one-dimensional Diophantine approximation, the Diophantine properties of a real number are characterized by its partial quotients, especially the growth of its large partial quotients. Notably, Kleinbock and Wadleigh [Proc. Amer. Math. Soc. 2018] made a seminal contribution by linking the improvability of Dirichlet's theorem to the growth of the product of consecutive partial quotients. In this paper, we extend the concept of Dirichlet non-improvable sets within the framework of shrinking target problems. Specifically, consider the dynamical system $([0,1), T)$ of continued fractions. Let $\{z_n\}_{n \ge 1}$ be a sequence of real numbers in $[0,1]$ and let $B > 1$. We determine the Hausdorff dimension of the following set: \[ \begin{split} \{x\in[0,1):|T^nx-z_n||T^{n+1}x-Tz_n|<B^{-n}\text{ infinitely often}\}. \end{split} \]