Related papers: Bifurcation sets arising from non-integer base exp…

Following in the footsteps of P. Erd\H{o}s and A. R\'enyi we compute the Hausdorff dimension of sets of numbers whose digits with respect to their $Q$-Cantor series expansions satisfy various statistical properties. In particular, we…

We initiate the study of the sets $H(c)$, $0<c<1$, of real $x$ for which the sequence $(kx)_{k\geq1}$ (viewed mod 1) consistently hits the interval $[0,c)$ at least as often as expected (i. e., with frequency $\geq c$). More formally, \[…

Let $\psi : \mathbb{R}_{>0}\rightarrow \mathbb{R}_{>0}$ be a non-increasing function. Denote by $W(\psi)$ the set of $\psi$-well-approximable points and by $E(\psi)$ the set of points $x\in[0,1]$ such that for any $0 < \epsilon < 1$ there…

The theory of uniform Diophantine approximation concerns the study of Dirichlet improvable numbers and the metrical aspect of this theory leads to the study of the product of consecutive partial quotients in continued fractions. It is known…

Given $x\in(0, 1]$, let $\mathcal U(x)$ be the set of bases $q\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^\infty d_i/q^i$. L\"{u}, Tan and Wu (2014) proved that $\mathcal U(x)$ is a…

For any field k and any integers m,n with 0 <= 2m <= n+1, let W_n be the k-vector space of sequences (x_0,...,x_n), and let H_m be the subset of W_n consisting of the sequences that satisfy a degree-m linear recursion, that is, for which…

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…

Let $b\geq3$ be an integer and $C(b,D)$ be the set of real numbers in $[0,1]$ whose $b$-ary expansion consists of digits restricted to a given set $D\subseteq\{0,\ldots,b-1\}$. Given an integer $t\geq2$ and a real, positive function $\psi$,…

We present a detailed Hausdorff dimension analysis of the set of real numbers where the product of consecutive partial quotients in their continued fraction expansion grow at a certain rate but the growth of the single partial quotient is…

We study the sets $\mathcal{L}$ and $\mathcal{M}\setminus\mathcal{L}$ near $3$, where $\mathcal{L}$ and $\mathcal{M}$ are the classical Lagrange and Markov spectra. More specifically, we construct a strictly decreasing sequence…

Let $q\in(1,2)$; it is known that each $x\in[0,1/(q-1)]$ has an expansion of the form $x=\sum_{n=1}^\infty a_nq^{-n}$ with $a_n\in\{0,1\}$. It was shown in \cite{EJK} that if $q<(\sqrt5+1)/2$, then each $x\in(0,1/(q-1))$ has a continuum of…

The graphs of Okamoto's functions, denoted by $K_q$, are self-affine fractal curves contained in $[0,1]^2$, parameterised by $q \in (1,2)$. In this paper we consider the cardinality and dimension of the intersection of these curves with…

Much has been written about expansions of real numbers in noninteger bases. Particularly, for a finite alphabet $\{0,1,\dots,\alpha\}$ and a real number (base) $1<\beta<\alpha+1$, the so-called {\em univoque set} of numbers which have a…

In this paper, we study the metrical theory of the growth rate of digits in L\"{u}roth expansions. More precisely, for $ x\in \left( 0,1 \right] $, let $ \left[ d_1\left( x \right) ,d_2\left( x \right) ,\cdots \right] $ denote the…

The escaping set I(f) of a transcendental meromorphic function f consists of all points which tend to infinity under iteration. The Eremenko-Lyubich class B consists of all transcendental meromorphic functions for which the set of finite…

For a positive integer $m$ let $\Omega _m=\{0,1, \cdots , m\}$ and \begin{align*} \mathcal B_2(m)=&\left \{q\in(1,m+1]: \text{$\exists\; x\in [0, m/(q-1)]$ has exactly }\right. \\ &\left. \text{two different $q$-expansions w.r.t. $\Omega…

For $\beta\in(1,2]$ the $\beta$-transformation $T_\beta: [0,1) \to [0,1)$ is defined by $T_\beta ( x) = \beta x \pmod 1$. For $t\in[0, 1)$ let $K_\beta(t)$ be the survivor set of $T_\beta$ with hole $(0,t)$ given by \[K_\beta(t):=\{x\in[0,…

We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we…

An open set U of the real numbers R is produced such that the expansion (R,+,x,U) of the real field by U defines a Borel isomorph of (R,+,x,N) but does not define N. It follows that (R,+,x,U) defines sets in every level of the projective…

Given a positive integer $M$ and $q \in (1, M+1]$ we consider expansions in base $q$ for real numbers $x \in \left[0, {M}/{q-1}\right]$ over the alphabet $\{0, \ldots, M\}$. In particular, we study some dynamical properties of the natural…