Uniform Diophantine approximation and run-length function in continued fractions

TL;DR

This paper investigates the multifractal properties of uniform and asymptotic approximation exponents in continued fractions, calculating Hausdorff dimensions of certain Diophantine sets and analyzing the multifractal nature of run-length functions.

Contribution

It provides new formulas for Hausdorff dimensions of uniform Diophantine approximation sets and explores their multifractal characteristics in continued fractions.

Findings

Calculated Hausdorff dimension of uniform Diophantine sets for algebraic irrationals.

Established multifractal properties of run-length functions in continued fractions.

Contributed to understanding uniform Diophantine approximation in number theory.

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 $\hat{\nu},$ we calculate the Hausdorff dimension of the uniform Diophantine set $$\mathcal{U}(y,\hat{\nu})=\Big\{x\in[0,1)\colon \forall N\gg1, \exists~ n\in[1,N], \text{ such that } |T^{n}(x)-y|<|I_{N}(y)|^{\hat{\nu}}\Big\}$$ 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.