For IFS generated by the first n branches of the Gauss map, lim (n→∞) (1 - H_n)/((1 - h_n) ln n) = 1, equivalently n(1 - H_n)/ln n → 6/π² by Hensley's result.
Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Ad- vanced Mathematics, vol
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Asymptotics of the Hausdorff measure for the Gauss map and its linearized analogue
For IFS generated by the first n branches of the Gauss map, lim (n→∞) (1 - H_n)/((1 - h_n) ln n) = 1, equivalently n(1 - H_n)/ln n → 6/π² by Hensley's result.