The Markoff and Lagrange Spectra on the Hecke Group H4

The Kim–Sim paper proves exactly the three claims: positive Hausdorff dimension immediately above the first accumulation point 2√2 (Theorem 1.1), two maximal gaps (√238/5, √10) and (√10, (2124√2+48√238)/1177) (Theorem 1.2), and a Hall ray (4√2, ∞) in the Lagrange spectrum (Theorem 1.3) . It also establishes the coding formulas expressing M(T) and L(T) as sup/lim sup over sections P*|Q of a doubly-infinite Romik sequence (Theorems 3.5–3.6) , the closedness of M(H4) and the inclusion L(H4) ⊂ M(H4) (Theorems 4.2–4.3) . By contrast, the model’s writeup mis-cites theorem numbers, asserts the lower gap endpoint √10 is realized by the periodic word (1232)∞ (the paper gives ∞(32)∞) and claims in general that both gap endpoints are attained by periodic H4-expansions, whereas the upper endpoint m0 is obtained via a specific two-sided sequence U = S*23232S (not periodic) with S = (31321312)∞ (Theorem 6.1) . The Hall-ray construction at 4√2 is correctly reported, but its explanation in the paper is via a tailored Cantor-set sum argument (Section 7) rather than by invoking external robustness results; see the explicit F+F computation and the proof of Theorem 1.3 . In sum: the paper’s arguments and conclusions are correct; the model’s account contains nontrivial factual inaccuracies and mis-citations despite capturing the main thrust.