On Dirichlet non-improvable numbers and shrinking target problems

2025/03/13 by Xiao, Qian
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.10381

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 ≥ 1} be a sequence of real numbers in [0,1] and let B &gt; 1. We determine the Hausdorff dimension of the following set: \[ \beginsplit \x∈[0,1):|Tⁿx-z_n||Tⁿ⁺¹x-Tz_n|<b></b>

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