On Dirichlet non-improvable numbers and shrinking target problems

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: $$\begin{array}{r}{\displaystyle \{x\in [0,1):\mathrm{\mid }{T}^{n}x-z_n\mathrm{\mid }\mathrm{\mid }{T}^{n+1}x-T{z}_n\mathrm{\mid }<B^{-n}\text{ infinitely often}\}\mathrm{.}}\end{array}$$