Abstract:
We consider the self-similar structure of the class of generalized Cantor sets $$\Gamma_{\mathcal{D}}=\Big\{\sum_{n=1}^\infty d_n\beta^{n}: d_n\in D_n, n\ge 1\Big\},$$ where $0<\beta<1$ and $D_n, n\ge 1,$ are nonempty and finite subsets of $\mathbb{Z}$. We give a necessary and sufficient condition for $\Gamma_{\mathcal{D}}$ to be a homogeneously generated self similar set. An application to the self-similarity of intersections of generalized Cantor sets will be given.

Abstract:
Given an integer $N\ge 2$ and a real number ${\beta}>1$, let $\Gamma_{{\beta},N}$ be the set of all $x=\sum_{i=1}^\infty {d_i}/{{\beta}^i}$ with $d_i\in\{0,1,\cdots,N-1\}$ for all $i\ge 1$. The infinite sequence $(d_i)$ is called a ${\beta}$-expansion of $x$. Let $\mathbf{U}_{{\beta},N}$ be the set of all $x$'s in $\Gamma_{{\beta},N}$ which have unique ${\beta}$-expansions. We give explicit formula of the Hausdorff dimension of $\mathbf{U}_{{\beta},N}$ for ${\beta}$ in any admissible interval $[{{\beta}}_L,{{\beta}}_U]$, where ${{\beta}_L}$ is a purely Parry number while ${{\beta}_U}$ is a transcendental number whose quasi-greedy expansion of $1$ is related to the classical Thue-Morse sequence. This allows us to calculate the Hausdorff dimension of $\U{N}$ for almost every $\beta>1$. In particular, this improves the main results of G{\'a}bor Kall{\'o}s (1999, 2001). Moreover, we find that the dimension function $f({\beta})=\dim_H\mathbf{U}_{{\beta},N}$ fluctuates frequently for ${\beta}\in(1,N)$.

Abstract:
Given a positive integer $M$ and a real number $q>1$, we consider the univoque set $\mathcal{U}_q$ of reals which have a unique $q$-expansion over the alphabet $\set{0,1,\cdots,M}$. In this paper we show that for any $x\in\mathcal{U}_q$ and all sufficiently small $\varepsilon>0$ the Hausdorff dimension $\dim_H\mathcal{U}_q\cap(x-\varepsilon, x+\varepsilon)$ equals either $\dim_H\mathcal{U}_q$ {or} zero. Moreover, we give a complete description of the typical points $x\in\mathcal{U}_q$ which satisfy \[ \dim_H\mathcal{U}_q\cap(x-\varepsilon, x+\varepsilon)=\dim_H\mathcal{U}_q\quad\textrm{for any}\quad \varepsilon>0, \] and prove that the set of typical points of $\mathcal{U}_q$ has full Hausdorff dimension. In particular, we show that if $\mathcal{U}_q$ is a Cantor set, then all points of $\mathcal{U}_q$ are typical points. This strengthen a result of de Vries and Komornik (Adv. Math., 2009).

Abstract:
We introduce a discrete time microscopic single particle model for kinetic transport. The kinetics is modeled by a two-state Markov chain, the transport by deterministic advection plus a random space step. The position of the particle after $n$ time steps is given by a random sum of space steps, where the size of the sum is given by a Markov binomial distribution (MBD). We prove that by letting the length of the time steps and the intensity of the switching between states tend to zero linearly, we obtain a random variable $S(t)$, which is closely connected to a well known (deterministic) PDE reactive transport model from the civil engineering literature. Our model explains (via bimodality of the MBD) the double peaking behavior of the concentration of the free part of solutes in the PDE model. Moreover, we show for instantaneous injection of the solute that the partial densities of the free and adsorbed part of the solute at time $t$ do exist, and satisfy the partial differential equations.

Abstract:
For a real number $q\in(1,2)$ and $x\in[0,1/(q-1)]$, the infinite sequence $(d_i)$ is called a \emph{$q$-expansion} of $x$ if $$ x=\sum_{i=1}^\infty\frac{d_i}{q^i},\quad d_i\in\{0,1\}\quad\textrm{for all}~ i\ge 1. $$ For $m=1, 2, \cdots$ or $\aleph_0$ we denote by $\mathcal{B}_m$ the set of $q\in(1,2)$ such that there exists $x\in[0,1/(q-1)]$ having exactly $m$ different $q$-expansions. It was shown by Sidorov (2009) that $q_2:=\min \mathcal{B}_2\approx1.71064$, and later asked by Baker (2015) whether $q_2\in\mathcal{B}_{\aleph_0}$? In this paper we provide a negative answer to this question and conclude that $\mathcal{B}_{\aleph_0} $ is not a closed set. In particular, we give a complete description of $x\in[0,1/(q_2-1)]$ having exactly two different $q_2$-expansions.

Abstract:
We study the shape of the probability mass function of the Markov binomial distribution, and give necessary and sufficient conditions for the probability mass function to be unimodal, bimodal or trimodal. These are useful to analyze the double-peaking results from a PDE reactive transport model from the engineering literature. Moreover, we give a closed form expression for the variance of the Markov binomial distribution, and expressions for the mean and the variance conditioned on the state at time $n$.

Abstract:
We consider a discrete time particle model for kinetic transport on the two dimensional integer lattice. The particle can move due to advection in the $x$-direction and due to dispersion. This happens when the particle is free, but it can also be adsorbed and then does not move. When the dispersion of the particle is modeled by simple random walk, strange phenomena occur. In the second half of the paper, we resolve these problems and give expressions for the shape of the plume consisting of many particles.

Abstract:
Let $\Gamma_{\beta,N}$ be the $N$-part homogeneous Cantor set with $\beta\in(1/(2N-1),1/N)$. Any string $(j_\ell)_{\ell=1}^\N$ with $j_\ell\in\{0,\pm 1,...,\pm(N-1)\}$ such that $t=\sum_{\ell=1}^\N j_\ell\beta^{\ell-1}(1-\beta)/(N-1)$ is called a code of $t$. Let $\mathcal{U}_{\beta,\pm N}$ be the set of $t\in[-1,1]$ having a unique code, and let $\mathcal{S}_{\beta,\pm N}$ be the set of $t\in\mathcal{U}_{\beta,\pm N}$ which make the intersection $\Gamma_{\beta,N}\cap(\Gamma_{\beta,N}+t)$ a self-similar set. We characterize the set $\mathcal{U}_{\beta,\pm N}$ in a geometrical and algebraical way, and give a sufficient and necessary condition for $t\in\mathcal{S}_{\beta,\pm N}$. Using techniques from beta-expansions, we show that there is a critical point $\beta_c\in(1/(2N-1),1/N)$, which is a transcendental number, such that $\mathcal{U}_{\beta,\pm N}$ has positive Hausdorff dimension if $\beta\in(1/(2N-1),\beta_c)$, and contains countably infinite many elements if $\beta\in(\beta_c,1/N)$. Moreover, there exists a second critical point $\alpha_c=\big[N+1-\sqrt{(N-1)(N+3)}\,\big]/2\in(1/(2N-1),\beta_c)$ such that $\mathcal{S}_{\beta,\pm N}$ has positive Hausdorff dimension if $\beta\in(1/(2N-1),\alpha_c)$, and contains countably infinite many elements if $\beta\in[\alpha_c,1/N)$.

Abstract:
Given two positive integers $M$ and $k$, let $\B_k$ be the set of bases $q>1$ such that there exists a real number $x$ having exactly $k$ different $q$-expansions over the alphabet $\{0,1,\cdots,M\}$. In this paper we investigate the smallest base $q_2$ of $\B_2$, and show that if $M=2m$ the smallest base $$q_2 =\frac{m+1+\sqrt{m^2+2m+5}}{2},$$ and if $M=2m-1$ the smallest base $q_2$ is the appropriate root of $$ x^4=(m-1)\,x^3+2 m\, x^2+m \,x+1. $$ Moreover, for $M=2$ we show that $q_2$ is also the smallest base of $\B_k$ for all $k\ge 3$. This turns out to be different from that for $M=1$.

Abstract:
We fix a positive integer $M$, and we consider expansions in arbitrary real bases $q>1$ over the alphabet $\{0,1,...,M\}$. We denote by $U_q$ the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension $D(q)$ of $U_q$ for each $q\in (1,\infty)$. Furthermore, we prove that the dimension function $D:(1,\infty)\to[0,1]$ is continuous, and has a bounded variation. Moreover, it has a Devil's staircase behavior in $(q',\infty)$, where $q'$ denotes the Komornik--Loreti constant: although $D(q)>D(q')$ for all $q>q'$, we have $D'<0$ a.e. in $(q',\infty)$. During the proofs we improve and generalize a theorem of Erd\H{o}s et al. on the existence of large blocks of zeros in $\beta$-expansions, and we determine for all $M$ the Lebesgue measure and the Hausdorff dimension of the set of bases in which $x=1$ has a unique expansion.