FRACTAL DIMENSIONS OF THE MARKOV AND LAGRANGE SPECTRA NEAR 3

The uploaded paper proves the asymptotic d(3+Y) = 2·W(e^{c0}|log Y|)/|log Y| + O(log|log Y|/|log Y|^2) with c0 = −log log((3+√5)/2) for Y→0+ (Theorem 1.2), and also shows an optimality statement for reasonable approximations (Theorem 1.3) . The candidate solution derives the same leading term and error rate via a different route (thermodynamic formalism for a renewal-type Gauss–Cantor set). The main caveat is that the candidate attributes a global identity d(t)=min{1,2D(t)} to Moreira; the paper only relies on equality of truncated dimensions dim_H(L∩(−∞,t))=dim_H(M∩(−∞,t)) and then develops bespoke coverings/renormalizations near 3 . This overstatement does not affect the final asymptotic near 3, where 2D(t)<1. Net: both reach the same result; the proofs are substantively different.