The packing measure of the linear Gauss system

Abstract

For every $k\in \mathbb{N}$, let $f_k:[\frac{1}{k+1},\frac{1}{k}]\to [0,1]$ be decreasing, linear functions such that $f_k(\frac{1}{k+1})=1$ and $f_k(\frac{1}{k})=0$, $k=1,2,\dots$. We define the iterated function system $S_n$ by limiting the collection of functions $f_k$ to first n, meaning $S_n=\{f_k{\}}_{k=1}^n$. Let $J_n$ denote the limit set of $S_n$. Then, $\underset{n\to \infty }{\lim }{\mathcal{P}}_{h_n}(J_n)=2$, where $h_n$ is the packing dimension of $J_n$ and ${\mathcal{P}}_{h_n}$ is the corresponding packing measure.