Monotone Tan–Zhou dimension conjecture for Lüroth approximation sets

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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 τ‾ψ\underline{\tau}_\psi denote the corresponding lower approximation exponent. Assume that ψ\psi is non-increasing. Monotone Tan–Zhou conjecture. Then

dim⁡L(ψ)=11+τ‾ψ.\dim L(\psi)=\frac{1}{1+\underline{\tau}_\psi}.

The monotonicity hypothesis is motivated by the paper's counterexamples to the unrestricted statement. Under this hypothesis, the asserted exact Hausdorff-dimension formula remains open in the supplied text.

References

Primary source

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

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