We define two types of the α-Farey maps Fα and $F_{\alpha, \flat}$ for $0 \lt \alpha \lt \tfrac{1}{2}$, which were previously defined only for $\tfrac{1}{2} \le \alpha \le 1$ by Natsui (2004). Then, for each $0 \lt \alpha \lt \tfrac{1}{2}$, we construct the natural extension maps on the plane and show that the natural extension of $F_{\alpha, \flat}$ is metrically isomorphic to the natural extension of the original Farey map. As an application, we show that the set of normal numbers associated with α-continued fractions does not vary by the choice of α, $0 \lt \alpha \lt 1$. This extends the result by Kraaikamp and Nakada (2000).

We study the freeness problem for multiplicative subgroups of $\operatorname{SL}_2(\mathbb{Q})$. For $q = r/p$ in $\mathbb{Q} \cap (0,4)$, where p is prime and $\gcd(r,p)=1$, we initiate the study of the algebraic structure of the group $\Delta_q$ generated by

\[A = \begin{pmatrix}1 & 0 \\ 1 & 1\end{pmatrix} \text{ and } Q_q = \begin{pmatrix}1 & q \\ 0 & 1\end{pmatrix}.\]

$\Delta_{r/p} = \overline{\Gamma}_1^{(p)}(r)$

$\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$

$r \leq 4$

$r = 5$

$r \in \{ p-1, p+1, (p+1)/2 \}$

$\Delta_{r/p}$

$J_2(r)$

$\operatorname{SL}_2(\mathbb{Z}[{1}/{p}])$

$J_2(r)$

Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension n are ergodic, resolving a conjecture of Messaoudi, Noguiera, and Schweiger [Ergodic properties of triangle partitions. Monatsh. Math. 157 (2009), 283–299]. This particular type of higher dimensional multi-dimensional continued fraction algorithm has recently been linked to the study of partition numbers, with the result that the underlying dynamics has combinatorial implications.

We establish explicit constructions of Mahler’s p-adic $U_{m}$-numbers by using Ruban p-adic continued fraction expansions of algebraic irrational p-adic numbers of degree m.

We show that there is a set $S \subseteq {\mathbb N}$ with lower density arbitrarily close to $1$ such that, for each sufficiently large real number $\alpha $, the inequality $|m\alpha -n| \geq 1$ holds for every pair $(m,n) \in S^2$. On the other hand, if $S \subseteq {\mathbb N}$ has density $1$, then, for each irrational $\alpha>0$ and any positive $\varepsilon $, there exist $m,n \in S$ for which $|m\alpha -n|<\varepsilon $.

Singular directions in a Veech surface are shown to be exactly the directions of its saddle connections.

For an (irreducible) recurrence equation with coefficients from $\mathbb Z[n]$ and its two linearly independent rational solutions $u_n,v_n$, the limit of $u_n/v_n$ as $n\to \infty $, when it exists, is called the Apéry limit. We give a construction that realises certain quotients of L-values of elliptic curves as Apéry limits.

This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set:

$$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$

where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$) shown by Hussain, Kleinbock, Wadleigh and Wang.

In the field of formal power series over a finite field, we prove a result which enables us to construct explicit examples of $U_{m}$-numbers by using continued fraction expansions of algebraic formal power series of degree $m>1$.

We consider the continued fraction expansion of real numbers under the action of a nonuniform lattice in $\text {PSL}(2,{\mathbb R})$ and prove metric relations between the convergents and a natural geometric notion of good approximations.

In this article we aim to investigate the Hausdorff dimension of the set of points $x \in [0,1)$ such that for any $r\in \mathbb {N}$,

$$ \begin{align*} a_{n+1}(x)a_{n+2}(x)\cdots a_{n+r}(x)\geq e^{\tau(x)(h(x)+\cdots+h(T^{n-1}(x)))} \end{align*} $$

$n\in \mathbb {N}$

$\tau $

$a_{n}(x)$

$r,\tau (x)$

$h(x)$

For a positive real number $t$, define the harmonic continued fraction

$$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr].\end{eqnarray}$$

$$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac{1}{t+6}-\cdots \,)}.\end{eqnarray}$$

Representations of the Cuntz algebra ${\mathcal{O}}_{N}$ are constructed from interval dynamical systems associated with slow continued fraction algorithms introduced by Giovanni Panti. Their irreducible decomposition formulas are characterized by using the modular group action on real numbers, as a generalization of results by Kawamura, Hayashi, and Lascu. Furthermore, a certain symmetry of such an interval dynamical system is interpreted as a covariant representation of the $C^{\ast }$-dynamical system of the “flip-flop” automorphism of ${\mathcal{O}}_{2}$.

Let $\unicode[STIX]{x1D6F9}:[1,\infty )\rightarrow \mathbb{R}_{+}$ be a non-decreasing function, $a_{n}(x)$ the $n$th partial quotient of $x$ and $q_{n}(x)$ the denominator of the $n$th convergent. The set of $\unicode[STIX]{x1D6F9}$-Dirichlet non-improvable numbers,

$$\begin{eqnarray}G(\unicode[STIX]{x1D6F9}):=\{x\in [0,1):a_{n}(x)a_{n+1}(x)>\unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$

$1/q^{2}\unicode[STIX]{x1D6F9}(q)$

${\mathcal{K}}(\unicode[STIX]{x1D6F9})$

${\mathcal{K}}(3\unicode[STIX]{x1D6F9})\subset G(\unicode[STIX]{x1D6F9})$

$s$

$s\in (0,1)$

$G(\unicode[STIX]{x1D6F9})\setminus {\mathcal{K}}(3\unicode[STIX]{x1D6F9})$

${\mathcal{K}}(\unicode[STIX]{x1D6F9})$

$G(\unicode[STIX]{x1D6F9})$

In this paper we extend and generalize, up to a natural bound of the method, our previous work Badziahin and Zorin [‘Thue–Morse constant is not badly approximable’, Int. Math. Res. Not. IMRN 19 (2015), 9618–9637] where we proved, among other things, that the Thue–Morse constant is not badly approximable. Here we consider Laurent series defined with infinite products $f_{d}(x)=\prod _{n=0}^{\infty }(1-x^{-d^{n}})$, $d\in \mathbb{N}$, $d\geq 2$, which generalize the generating function $f_{2}(x)$ of the Thue–Morse number, and study their continued fraction expansion. In particular, we show that the convergents of $x^{-d+1}f_{d}(x)$ have a regular structure. We also address the question of whether the corresponding Mahler numbers $f_{d}(a)\in \mathbb{R}$, $a,d\in \mathbb{N}$, $a,d\geq 2$, are badly approximable.

In this paper, we study rational approximations for certain algebraic power series over a finite field. We obtain results for irrational elements of strictly positive degree satisfying an equation of the type

$$\alpha =\frac{A{{\alpha }^{q}}+B}{C{{\alpha }^{q}}},$$

where $\left( A,B,C \right)\,\in \,{{\left( {{\mathbb{F}}_{q}}\left[ X \right] \right)}^{2}}\times \mathbb{F}_{q}^{*}\left[ X \right]$ . In particular, under some conditions on the polynomials $A,\,B$ and $C$ , we will give well approximated elements satisfying this equation.

Denote the nth convergent of the continued fraction α=[a0;a1,a2,…] by pn/qn=[a0;a1,…,an]. In this paper we give exact formulae for the quantities Dn:=qnα−pn in several typical types of Tasoev continued fractions. A simple example of the type of Tasoev continued fraction considered is α=[0;ua,ua2,ua3,…].

We establish measures of non-quadraticity and transcendence measures for real numbers whose sequence of partial quotients has sublinear block complexity. The main new ingredient is an improvement of Liouville’s inequality giving a lower bound for the distance between two distinct quadratic real numbers. Furthermore, we discuss the gap between Mahler’s exponent w2 and Koksma’s exponent w*2.

We give a two dimensional extension of the three distance theorem. Let $\theta $ be in ${{\mathbf{R}}^{2}}$ and let $q$ be in $\mathbf{N}$ . There exists a triangulation of ${{\mathbf{R}}^{2}}$ invariant by ${{\mathbf{Z}}^{2}}$ -translations, whose set of vertices is ${{\mathbf{Z}}^{2}}\,+\,\{0,\,\theta ,\,\ldots ,\,q\theta \}$ , and whose number of different triangles, up to translations, is bounded above by a constant which does not depend on $\theta $ and $q$ .