Lüroth Expansions in Diophantine Approximation: Metric Properties and Conjectures
Ying Wai Lee
TL;DR
The paper develops a Lüroth-specific metric theory for well-approximable numbers, defining $L(\psi)$ and $L(\tau)$ using Lüroth convergents and proving Khintchine-type, Jarnik–Besicovitch-type, and Dodson analogue results. It establishes a complete Khintchine-type criterion via $\sum_q -\psi(q)\log\psi(q)/q$ with a supplementary proof, determines the exact Hausdorff dimension $\dim L(\tau)=1/(1+\tau)$ for all $\tau\ge0$ (and that $\mathcal H^{1/(1+\tau)}(L(\tau))=\infty$) through the Beresnevich–Velani Mass Transference Principle, and extends dimension results to general $\psi$ through lower/upper orders and refined bounds. The work also provides a counterexample showing monotonicity cannot be dropped from conjectures, and offers a partial result toward a revised conjecture under additional hypotheses. Together, these results advance metric Diophantine approximation in the Lüroth setting, clarify the role of monotonicity, and connect mass transference principles to fractal dimensions of Lüroth-convergent sets.
Abstract
This paper focuses on the metric properties of Lüroth well approximable numbers, studying analogous of classical results, namely the Khintchine Theorem, the Jarník--Besicovitch Theorem, and the result of Dodson. A supplementary proof is provided for a measure-theoretic statement originally proposed by Tan--Zhou. The Beresnevich--Velani Mass Transference Principle is applied to extend a dimensional result of Cao--Wu--Zhang. A counterexample is constructed, leading to a revision of a conjecture by Tan--Zhou concerning dimension, along with a partial result.