Uniform Diophantine approximation and run-length function in continued fractions

TL;DR

The paper investigates the multifractal structure of uniform and asymptotic Diophantine exponents in continued fractions under the Gauss map. It derives explicit Hausdorff-dimension formulas for the uniform-target sets $\mathcal{U}(y,\hat{\nu})$, the level sets $E(\hat{\nu},\nu)$, and the run-length related sets $F(\alpha)$ and $G(\beta)$, including their intersections. The authors develop and employ pressure-function techniques and pre-dimensional numbers to obtain matching upper and lower bounds, constructing Cantor-type subsets and dimension-preserving maps to prove exact dimensions. The results illuminate the multifractal geometry of uniform and asymptotic Diophantine approximation in the continued fraction setting and connect to the multifractal properties of the run-length function. Overall, the work provides precise dimension formulas and a unified framework for understanding uniform approximation in dynamical systems arising from continued fractions.

Abstract

We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, %given a nonnegative reals $\hatν,$ we calculate the Hausdorff dimension of the uniform Diophantine set $$\mathcal{U}(y,\hatν)=\Big\{x\in[0,1)\colon \forall N\gg1, \exists~ n\in[1,N], \text{ such that } |T^{n}(x)-y|<|I_{N}(y)|^{\hatν}\Big\}$$ for algebraic irrational points $y\in[0,1)$. These results contribute to the study of the uniform Diophantine approximation, and apply to investigating the multifractal properties of run-length function in continued fractions.

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1: K
• proof
• Lemma 2.2: K
• Lemma 2.3: K