Tan–Zhou's Hausdorff-dimension conjecture for Lüroth approximation sets

Let L(ψ)L(\psi) be the set of points defined by the Lüroth approximation condition associated with a function ψ:N(0,1]\psi:\mathbb{N}\to(0,1], and let τψ\underline{\tau}_\psi denote the corresponding lower approximation exponent. Tan–Zhou's conjecture. For every function ψ:N(0,1]\psi:\mathbb{N}\to(0,1],

dimL(ψ)=11+τψ.\dim L(\psi)=\frac{1}{1+\underline{\tau}_\psi}.

This is proposed as an analogue of a result of Dodson. The statement is false without a monotonicity assumption, as the paper gives examples with the same value of τψ\underline{\tau}_\psi but Hausdorff dimension zero; the monotone version is therefore the surviving open conjecture.

Sources & referencesView supporting material

Primary source

Ying Wai Lee, “Lüroth Expansions in Diophantine Approximation: Metric Properties and Conjectures”, arXiv:2502.08408 (2025).

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