Dichotomy for the Hausdorff dimension of nonergodic directions on translation surfaces

The uploaded paper proves exactly the general dichotomy the model labeled “likely open.” Its Theorem 1.1 states: for any irrational (λ, µ) and for any norm on R^2, if {q_k} are the denominators of best approximation vectors of (λ, µ), then HDim NE(X_{λ,µ}, ω) ∈ {0, 1/2}, with HDim = 1/2 if and only if ∑_k (log log q_{k+1})/q_k < ∞ . The paper further derives that the Pérez–Marco condition is norm-independent across all norms (Corollary 1.3) . The dimension-0 half is proved in Section 3 via Z-expansions: under divergence of the series, every nonergodic direction is shown to be Liouville relative to Z, yielding HDim = 0 (Theorem 3.10) , with uniform best-approximation estimates from Cheung (Theorems 2.6–2.7) used to make the argument norm-agnostic . The dimension-1/2 half is established by constructing a tree of slits (Diophantine and Liouville constructions), verifying the summable cross-products criterion (Theorem 2.2) to place the Cantor-type set in NE, and applying a Falconer-type lower bound; see the outline and culminating lemmas (Lemmas 6.9, 6.10, and the special-case finishing Lemma 7.6) . In sum, the paper delivers a complete if-and-only-if criterion for all irrational (λ, µ) and all norms, directly contradicting the model’s claim that the general statement was not in the literature as of 2025-07-16.