2025/11/04 by Tan, Bo, Tian, Chen, Wang, Baowei +1
Mathematics · #Mathematical Dynamics and Fractals #Fixed Point Theorems Analysis #Limits and Structures in Graph Theory
paper · doi:10.48550/arxiv.2511.02492
Let (X, d) be a compact metric space, and let Q ⊂ X be countable. Given functions R: Q → ℝ+ and ϕ: ℝ+ → ℝ+, we consider the set E(Q, R, ϕ) of points x ∈ X that ``hit'' the shrinking balls B(ξ,ϕ(R(ξ))) for infinitely many ξ∈ Q, yet, for every ε∈ (0,1), are eventually ``cleared out'' from the slightly smaller neighborhoods B(ξ,(1-ε)ϕ(R(ξ))), that is, they lie outside all but finitely many of these smaller balls. We give sufficient conditions (also necessary under mild assumptions) for E(Q, R, ϕ) to have infinite Hausdorff f-measure. This setting generalizes both the classical set Exact(ψ) of exactly ψ-approximable points (with ψ non-increasing) and certain types of restricted Diophantine approximation sets.