On Dirichlet non-improvable numbers and shrinking target problems
Qian Xiao
TL;DR
This work extends Dirichlet non-improvable sets to shrinking target problems in one-dimensional Diophantine approximation by analyzing the Gauss-map dynamics of continued fractions. It defines the shrinking-target set $E_2(\{z_n\}_{n\ge 1},B)$ via the product $|T^n x-z_n|\,|T^{n+1}x-Tz_n|<B^{-n}$ and proves that its Hausdorff dimension equals $s^*=\\limsup_{n\to\infty} s_n$, with $s_n$ determined by pre-dimension numbers $s_{n,1},s_{n,2},s_{n,3}$. The proof combines an upper-bound argument based on delicate cylinder-cover estimates and three potential-driven pressure function analyses with a complementary lower-bound construction using mass distribution across three cases where $s^*$ is attained by one of the $s_{n,i}$. The results generalize Kleinbock–Wadleigh’s link between improvability and partial-quotient growth to shrinking-target settings, enriching the shrinking-target literature for continued fractions and Diophantine approximation.
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: \[ \begin{split} \{x\in[0,1):|T^nx-z_n||T^{n+1}x-Tz_n|<B^{-n}\text{ infinitely often}\}. \end{split} \]