Asymptotics of the Hausdorff measure for the Gauss map and its linearized analogue

For $n\in\mathbb N$ we consider the set $J_n$ of points in the interval [0,1] whose continued fraction expansion entries are bounded by n. Similarly, we consider the set $J_n$ for the linearized analogue of the Gauss map. We study the asymptotic of the Hausdorff measure of the set $J_n$, (evaluated at its Hausdorff dimension). We obtain precise asymptotics for the linearized Gauss map and the same one -sided bound for the asymptotics for the original Gauss map.