Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

The paper (Pollicott–Vytnova) proves the five window bounds and the global estimate dim_H(M \ L) < 0.882220 by combining Matheus–Moreira’s structural reductions with a validated transfer-operator computation, explicitly stated as Theorem 1.3 with the numerical values 0.731541, 0.855228, 0.872773, 0.812238, 0.882220 and the global bound 0.882220 . Their Section 4.1 explains the reliance on [31] for the inclusions/forbidden-word models and implements a rigorous bisection/interpolation scheme to certify dimensions of the relevant IFS limit sets . The model’s solution reproduces the same strategy: (i) symbolic coding of M and L, (ii) use of Matheus–Moreira’s gap principle and SFT reductions on specified windows, (iii) covering (M \ L) ∩ I by sums of Gauss–Cantor sets and applying dim_H(A+B) ≤ dim_H A + dim_H B, and (iv) importing the PV certified dimensions to obtain the stated bounds, hence the same final inequality. Differences are expository (sumset language versus the paper’s ‘dim(X_M)+constant’ presentation), not substantive. The numbers and logic agree with the paper’s results (see the worked parts for the intervals (√13,3.84), (3.84,3.92), and (3.92,4.01) in §4.1.1) ; the Hall–Freiman ray and the location of c_F used for the global conclusion are also recalled in the paper’s introduction .