Extend the closed-set characterisation (Theorem 3.5) to Gδ sets

Establish whether Theorem 3.5 extends to arbitrary Gδ subsets of S^{n−1}: specifically, determine if for every non-empty Gδ set X ⊂ S^{n−1} there exists a discrete set E ⊂ D^{n} that is separated and well-approximated such that the radial limit set Lrad(E) equals X and the identity dim_H(X) = min{δ(E) : Lrad(E) = X} holds, where δ(E) = inf{s > 0 : ∑_{x∈E} (1−|x|)^s < ∞} and Lrad(E) = ⋃_{c≥1} {z ∈ S^{n−1} : ∀ r > 0, ∃ x ∈ E with |x − z| < c(1−|x|) ≤ cr}.

Background

The paper proves a sharp characterisation (Theorem 3.5) for closed sets X ⊂ S^{n−1}, showing dim_H(X) equals the minimum critical exponent over discrete sets E whose radial limit set equals X, within the class of separated and well-approximated E.

Radial limit sets Lrad(E) are shown to be Gδ sets (Proposition 3.6), motivating an extension of the closed-set result to the natural class of Gδ sets that arise as radial limit sets. The authors state they do not know how to prove this extension and instead provide a partial result for Fσ sets (Theorem 3.7), which only achieves inclusion Lrad(E) ⊇ X.