arXiv subjects

Rafał Tryniecki

Publications and source records attributed to Rafał Tryniecki.

3 recordsLinked to original sources

arXiv2026

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

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 $g_k$, $k\in\mathbb N$, 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 $$ \lim_{n\to\infty} \frac{1-H_n}{(1-h_n)\ln n}= 1 $$ and equivalently, due to Hensley's result, $$ \lim_{n\to\infty} \frac{n(1-H_n)}{\ln n}= \frac{6}{π^2}, $$ 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 $J_n$, $H_n:=H_{h_n}(J_n)$.

math.DS↗

arXiv2026

On the conditions for the continuity of the Hausdorff measure

Let $(b_k)_{k = 0}^\infty$ be strictly decreasing sequence of real numbers such that $b_0 = 1$ and $\{f_k:[b_k,b_{k-1}]\to [0,1]\}_{k\in\N}$ be decreasing functions such that $f_k(b_k) = 1$ and $f_k(b_{k-1}) = 0$, $k = 1, 2, \dots$. By $g_k: [0,1] \to [b_k, b_{k-1}]$ we denote the inverse of $f_k$ for $k = 1,2 \dots$. First, we define iterated function system (IFS) $S_n$ by limiting the collection of functions $g_k$ to first n, meaning $S_n = \{g_k \}_{\{k=1, \dots n\}}$. Let $J_n$ denote the limit set of $S_n$. In the first part, we show that if $S_n$ fulfills the following two conditions: (1)~$\lim\limits_{n \to \infty} \left(1-h_n\right) \ln{n} = 0 $ where $h_n$ is the Hausdorff dimension of $J_n$, and (2)~$\sup \limits_{k\in \mathbb{N}} \left \{\frac{b_k-b_{k+1}}{b_{k+1}} \right \} < \infty $, then $\lim\limits_{n\to \infty} H_{h_n}(J_n) = 1 = H_1(J)$, where $h_n$ is the Hausdorff dimension of $J_n$ and $H_{h_n}$ is the corresponding Hausdorff measure. In the second part, we provide four conditions for IFS consisting of nonlinear functions $f_k$ which guarantee that $\lim\limits_{n\to \infty} H_{h_n}(J_n) = 1 = H_1(J)$, where $h_n$ is the Hausdorff dimension of $J_n$ and $H_{h_n}$ is the corresponding Hausdorff measure. We also provide a wide collection of examples of families of IFSes fulfilling those assumptions.

math.DS↗

arXiv2024

Packing measure of the linear Gauss system

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 iterated function system (IFS) $S_n$ 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 $\lim\limits_{n\to \infty} \mathcal{P}_{h_n}(J_n) = 2$, where $h_n$ is the packing dimension of $J_n$ and $\mathcal{P}_{h_n}$ is the corresponding packing measure.

math.DS↗