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On Dirichlet non-improvable numbers and shrinking target problems

Part of: Classical measure theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  11 June 2025

QIAN XIAO
QIAN XIAO*
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, P.R. China
*
e-mail: xiaoqianmath@163.com
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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. 146(5) (2018), 1833–1844] 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: $ \{x\in [0,1):|T^nx-z_n||T^{n+1}x-Tz_n|<B^{-n}\text { infinitely often}\}. $

Keywords

Dirichlet non-improvable setsshrinking target problemsHausdorff dimension

MSC classification

Primary: 11K50: Metric theory of continued fractions
Secondary: 28A80: Fractals
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , First View , pp. 1 - 29
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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