METRICAL THEOREMS FOR UNCONVENTIONAL HEIGHT FUNCTIONS

The paper proves: (a) for almost every x in R^d, ω_{H_max}(x)=ω_{H_{prod^{1/d}}}(x)=ω_{H_min}(x)=2 (Theorem 1.4), using continued fractions; (b) for Θ∈{max, prod^{1/d}} and τ≥2, dim_H S_Θ(τ)=2d/τ (Theorem 1.5), via Mass Transference and a Hausdorff–Cantelli upper bound; and (c) for monotone ψ, S_d(ψ,min)=⋃_{i=1}^d(R^{i-1}×S_1(ψ)×R^{d-i}) (Theorem 1.6) . The candidate solution reaches the same conclusions. Methods differ: the model uses a Borel–Cantelli/ubiquity route for (a) and Mass Transference for (b), while the paper’s (a) argument is by continued fractions. One minor issue in the model’s (a)/(b) “max-height” counting omits a factor of k^{-1} in the shell counting, leading to a summability exponent off by 1; this is easily corrected and does not alter the thresholds τ=2 and s=2d/τ. Overall, statements align and proofs are reconcilable.