Related papers: Periodic unique codings of fat Sierpinski gasket

Given $\beta\in(1,2)$ the fat Sierpinski gasket $\mathcal S_\beta$ is the self-similar set in $\mathbb R^2$ generated by the iterated function system (IFS) \[ f_{\beta,d}(x)=\frac{x+d}{\beta},\quad d\in\mathcal A:=\{(0, 0), (1,0), (0,1)\}.…

Given an integer $M\ge 1$ and $\beta\in(1, M+1)$, let $S_{\beta, M}$ be the fat Sierpinski gasket in $\mathbb R^2$ generated by the iterated function system $\left\{f_d(x)=\frac{x+d}{\beta}: d\in\Omega_M\right\}$, where…

Let $\beta\in(1,2)$. Each $x\in[0,\frac{1}{\beta-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty \epsilon_k\beta^{-k}, \] where $\epsilon_k\in\{0,1\}$ for all $k$ (a $\beta$-expansion of $x$). If $\beta>\frac{1+\sqrt5}{2}$, then,…

We consider the iterated function system (IFS) $$f_{\vec{q}}(\vec{z})=\frac{\vec{z}+\vec{q}}{\beta},\vec{q}\in\{(0,0),(1,0),(0,1)\}.$$ As is well known, for $\beta = 2$ the attractor, $S_\beta$, is a fractal called the Sierpi\'nski…

For $\beta\in(1,2]$ let $T_\beta: [0,1)\to[0,1); x\mapsto \beta x\pmod 1$. In this paper we study the periodic points in the open dynamical system $([0,1), T_\beta)$ with a hole $[0,t)$. For $p\in\mathbb{N}$ we characterize the largest $t$,…

In this paper we consider a general class $\mathcal E$ of self-similar sets with complete overlaps. Given a self-similar iterated function system $\Phi=(E, \{f_i\}_{i=1}^m)\in\mathcal E$ on the real line, for each point $x\in E$ we can find…

S. Baker (2019), B. B\'ar\'any and A. K\"{a}enm\"{a}ki (2019) independently showed that there exist iterated function systems without exact overlaps and there are super-exponentially close cylinders at all small levels. We adapt the method…

An identifying code in a graph is a subset of vertices having a nonempty and distinct intersection with the closed neighborhood of every vertex. We prove that the infimum density of any identifying code in $S_k$ (an infinite strip of $k$…

Let $F=\{\mathbf{p}_0,\ldots,\mathbf{p}_n\}$ be a collection of points in $\mathbb{R}^d.$ The set $F$ naturally gives rise to a family of iterated function systems consisting of contractions of the form $$S_i(\mathbf{x})=\lambda \mathbf{x}…

In this article, we study the smallest gaps of the log-gas $\beta$-ensemble on the unit circle (C$\beta$E), where $\beta$ is any positive integer. The main result is that the smallest gaps, after being normalized by $n^{\frac…

We consider numbers of the form $S_\beta(\boldsymbol{u}):=\sum_{n=0}^\infty \frac{u_n}{\beta^n}$ for $\boldsymbol{u}=\langle u_n \rangle_{n=0}^\infty$ a Sturmian sequence over a binary alphabet and $\beta$ an algebraic number with…

For a (non-unit) Pisot number $\beta$, several collections of tiles are associated with $\beta$-numeration. This includes an aperiodic and a periodic one made of Rauzy fractals, a periodic one induced by the natural extension of the…

$M$ is a cpt. Riemannian manifold without boundary, $f\in\mathrm{Diff}^{1+\beta}(M)$. In [Sarig13], for all $\chi>0$, for every small enough $\epsilon>0$, Sarig had first constructed a coding $\widehat{\pi}:\widehat{\Sigma}\rightarrow M$…

The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. We study the number of independent sets $m_{d,b}(n)$ on the generalized Sierpinski…

Let $\Omega$ be a sufficiently regular bounded open connected subset of $\mathbb{R}^n$ such that $0 \in \Omega$ and that $\mathbb{R}^n \setminus \mathrm{cl}\Omega$ is connected. Then we take $(q_{11},\dots, q_{nn})\in ]0,+\infty[^n$ and $p…

We consider a sufficiently regular bounded open connected subset $\Omega$ of $\mathbb{R}^n$ such that $0 \in \Omega$ and such that $\mathbb{R}^n \setminus \cl\Omega$ is connected. Then we choose a point $w \in ]0,1[^n$. If $\epsilon$ is a…

In this work, we introduce $\beta$-expansion, a notion borrowed from number theory, as a theoretical framework to study fast construction of polar codes based on a recursive structure of universal partial order (UPO) and polarization weight…

Two sets $A, B \subseteq \{0, 1\}^n$ form a Uniquely Decodable Code Pair (UDCP) if every pair $a \in A$, $b \in B$ yields a distinct sum $a+b$, where the addition is over $\mathbb{Z}^n$. We show that every UDCP $A, B$, with $|A| =…

We study long-range percolation on the hierarchical lattice of order $N$, where any edge of length $k$ is present with probability $p_k=1-\exp(-\beta^{-k} \alpha)$, independently of all other edges. For fixed $\beta$, we show that the…

In 1960, Sierpi\'nski proved that there exist infinitely many odd positive integers $k$ such that $k\cdot 2^n+1$ is composite for all positive integers $n$. In this paper, we prove some generalizations of Sierpi\'nski's theorem with $2^n$…