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

Rafał Tryniecki; Mariusz Urbański; Anna Zdunik

paper

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

arXiv:2504.02135

Abstract

Let $G (x) := {1/ x}$ be the Gauss map. By $g_{n} (x) = \frac{1}{x + n}$ we denote its continuous/real analytic inverse branches. We define iterated function system (IFS) $G_{n}$ by limiting the collection of functions $g_{k}$ , $k \in N$ , to the first $n$ elements, meaning that $G_{n} = {g_{k}}_{k = 1}^{n}$ . We are interested in the asymptotics of the Hausdorff measure of the limit set $J_{n}$ i. e. set consisting of irrational elements of $[0, 1]$ having continued fraction expansion with entries at most $n$ . In the first part of the paper, we deal with the piecewise-linear analogue of the Gauss map and resulting IFSs. We prove that \[ \lim \limits_{n \to \infty } \frac{1-H_n(J_n)}{1-h_n} \cdot \frac{1}{\ln n} = 1, \] where $J_{n}$ is the limit set of the piecewise-linear analogue of $G_{n}$ , $h_{n}$ is its Hausdorff dimension and $H_{n}$ is the value of $h_{n}$ -dimensional Hausdorff measure of the set $J_{n}$ , $H_{n} := H_{h_{n}} (J_{n})$ . In the second part, we focus on the IFS generated by the first $n$ branches of Gauss map and prove, as our main result, that $n \to \infty lim \frac{1 - H _{n}}{( 1 - h _{n} ) ln n} = 1$ and equivalently, due to Hensley's result, $n \to \infty lim \frac{n ( 1 - H _{n} )}{ln n} = \frac{6}{π ^{2}},$ where $J_{n}$ is the limit set of the system $G_{n}$ , i.e. the set consisting of irrational numbers in $[0, 1]$ that continued fraction expansion with entries not exceeding $n$ . Similarly as for the piecewise linear map, $h_{n}$ is the Hausdorff dimension of $J_{n}$ and $H_{n}$ is the value of $h_{n}$ -dimensional Hausdorff measure of the set $J_{n}$ , $H_{n} := H_{h_{n}} (J_{n})$ .