Exploring simultaneous characterizations of the generalized local fractal dimension functions

For x∈ Rⁿ and F⊆ Rⁿ, we introduce the generalized local Hausdorff dimension function of F at x and the generalized local packing dimension function of F at x as Df,gH,loc(x,F) = limr→ 0 Df,gH(Fᵣ(x)) andDf,gP,loc(x,F) = limr→ 0 Df,gP(Fᵣ(x)),respectively, where Df,gH is the generalized Hausdorff dimension, Df,gP is the generalized packing dimension, and Bᵣ(x) is the ball of center x∈ Rⁿ and radius $ r>0 $. Next, we provide a thorough description of the set of functions that serve as generalized local Hausdorff dimension functions and a complete characterization of functions that are generalized local Hausdorff dimension functions for generalized Hausdorff dimension and generalized local packing dimension functions for generalized packing dimension. Finally, we present a brief and straightforward demonstration indicating that any pair of continuous functions φ, ψ defined on Rⁿ and bounded between 0 and Df,g_ H( Rⁿ) such that φ≤ψ and under some assumptions, it is feasible to select a set F that concurrently serves as the generalized local Hausdorff dimension function for φ and the generalized local packing dimension function for ψ.