Related papers: Range-Renewal Structure in Continued Fractions

A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers, when represented using continued fractions, exhibit partial quotients that grow at specific rates. For any positive function…

Let $f\in \mathbb{R}[x_1,\ldots, x_k]$, for $k\ge 2$. For any finite sets $A_1,\ldots, A_k\subset \mathbb{R}$, consider the set $$ f(A_1,\ldots, A_k):=\{f(a_1,\ldots, a_k)\mid (a_1,\cdots,a_k)\in A_1\times\cdots \times A_k\}, $$ that is,…

We show a correspondence between simple continued fraction expansions of irrational numbers and irreducible permutative representations of the Cuntz algebra ${\cal O}_{\infty}$. With respect to the correspondence, it is shown that the…

We introduce a family of maps generating continued fractions where the digit $1$ in the numerator is replaced cyclically by some given non-negative integers $(N_1,\ldots,N_m)$. We prove the convergence of the given algorithm, and study the…

We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This latter condition requires that a sequence can be…

A well known theorem of Lagrange states that the simple continued fraction of a real number $\alpha$ is periodic if and only if $\alpha$ is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two…

An Engel series is a sum of the reciprocals of an increasing sequence of positive integers, which is such that each term is divisible by the previous one. Here we consider a particular class of Engel series, for which each term of the…

We consider a sequence of random variables $(R_n)$ defined by the recurrence $R_n=Q_n+M_nR_{n-1}$, $n\ge1$, where $R_0$ is arbitrary and $(Q_n,M_n)$, $n\ge1$, are i.i.d. copies of a two-dimensional random vector $(Q,M)$, and $(Q_n,M_n)$ is…

We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Numerous distinct such expansions are possible for a complex number. They can be arrived at through various algorithms, as…

Let $\omega$ be a sequence of positive integers. Given a positive integer $n$, we define $$ r_n(\omega) = | \{ (a,b)\in \mathbb{N}\times \mathbb{N}\colon a,b \in \omega, a+b = n, 0 <a<b \}|. $$ S. Sidon conjectured that there exists a…

We present modified proof of a certain version of Kolmogorov's strong law of large numbers for calculation of Lebesgue Integrals by using uniformly distributed sequences in $(0,1)$. We extend the result of C. Baxa and J. Schoi$\beta$engeier…

This paper investigates the quadratic irrationals that arise as periodic points of the Gauss type shift associated to the odd continued fraction expansion. It is shown that these numbers, which we call O-reduced, when ordered by the length…

Let $r_1,\ldots,r_s:\mathbb{Z}_{n\geqslant 0}\to\mathbb{C}$ be linearly recurrent sequences whose associated eigenvalues have arguments in $\pi\mathbb{Q}$ and let $F(z):=\sum_{n\geqslant 0}f(n)z^n$, where $f(n)\in\{r_1(n),\ldots,$…

We prove a continued fraction expansion for the reciprocal of a certain $q$-series. All the specialists in the world are asked whether it is new or not.

Let $s(n)$ be the number of different remainders $n \bmod k$, where $1 \leq k \leq \lfloor n/2 \rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has…

Let $\{a_n(x)\}_{n\geq1}$ be the sequence of digits of $x\in(0,1)$ in infinite iterated function systems with polynomial decay of the derivative. We first study the multifractal spectrum of the convergence exponent defined by the sequence…

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,\ldots,d_{k}],$ with all partial quotients…

For a given quadratic irrational $\alpha$, let us denote by $D(\alpha)$ the length of the periodic part of the continued fraction expansion of $\alpha$. We prove that for a positive integer $d$, which is not a perfect square, the sequence…

We prove a quantitative distortion theorem for iterated function systems that generate sets of continued fractions. As a consequence, we obtain upper and lower bounds on the Hausdorff dimension of any set of real or complex continued…

Let $k$ be a perfect complete valued field with a nontrivial non-archimedean norm $|\cdot|$ and $\omega\in k$ with $0<|\omega|<1.$ Let $X$ be a reduced and normal $k$-analytic space. Then $O^{\circ}\simeq…