L\"uroth Expansions in Diophantine Approximation: Metric Properties and Conjectures

TL;DR

This paper investigates the metric properties of Lüroth expansions in Diophantine approximation, extending classical theorems, providing new proofs, and revising existing conjectures through measure and dimension analysis.

Contribution

It extends classical Diophantine approximation results to Lüroth expansions, offers new measure-theoretic proofs, and revises a conjecture based on counterexamples and partial results.

Findings

Extended Khintchine and Jarnik-Besicovitch theorems to Lüroth expansions

Provided a supplementary proof for Tan-Zhou's measure-theoretic statement

Constructed a counterexample leading to a revised conjecture on dimension

Abstract

This paper focuses on the metric properties of L\"uroth well approximable numbers, studying analogous of classical results, namely the Khintchine Theorem, the Jarn\'ik--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.