Remove log α(n) term from ε-additive Steiner spanner bounds in H^d

Improve the ε-additive Steiner spanner construction in d-dimensional hyperbolic space H^d by eliminating the log α(n) factor from the current edge bound O_d(ε^{(1−d)/2}·log(α(n)/ε)·n), or prove a matching lower bound that shows this dependence is necessary.

Background

The authors present an ε-additive Steiner spanner in H^d with near-linear size, where the edge count includes a log α(n) term stemming from the use of transitive closure spanners on certain trees. Although this term is extremely slowly growing, it prevents achieving purely near-linear dependence in n without additional polylogarithmic factors.

They explicitly highlight that reducing the edge count by removing the log α(n) term is an open question, hinting at potential improvements either by new geometric constructions or by sharper combinatorial structures that avoid the current dependency.