Lüroth Expansions in Diophantine Approximation: Metric Properties and Conjectures

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.