An Improved Convergence Case for Diophantine Approximations on IFS Fractals

The paper proves the convergence part of a Khinchine-type theorem on self-similar IFS fractals for all exponents 0 < α < ϖ, where ϖ = min_l α_l(μ_K)(d−l+1), by importing a contraction hypothesis from Khalil and using random-walk/homogeneous-dynamics tools plus the Dani correspondence. The candidate solution only reaches the Pollington–Velani hyperplane threshold α = α_1(μ_K) via the simplex-lemma/absolute-decay route and explicitly cannot upgrade to all α < ϖ. The paper’s argument is coherent: Theorem 2 establishes the needed contraction with exponent α = (ρϖ)/(d+1) (ρ < 1), yielding exponentially recurrent sets; excursion bounds translate to bounds for the minimal vector along the flow; the Dani correspondence plus a change of variables yields the stated summability criterion and conclusion μ_K(ψ-approximable) = 0. While the paper treats Theorem 2 as a black-box import from [Kha20], the remainder is rigorous and correctly assembled. Hence the paper’s result stands, and the candidate’s is strictly weaker.