Related papers: Continued Fractions with Multiple Limits

Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite subset S of P such that the statistics of the period of the continued fraction expansions along the sequence {px: p\in S} approach…

A partition of the positive integers into sets $A$ and $B$ {\em avoids} a set $S\subset\N$ if no two distinct elements in the same part have a sum in $S$. If the partition is unique, $S$ is {\em uniquely avoidable.} For any irrational…

We give an exponential upper bound on the probabilitywith which the denominator of the $n$th convergent in the regular continued fraction expansion stays away from the mean $\frac{n\pi^2}{12\log2}$. The exponential rate is best possible,…

For a large integer $m,$ we obtain an asymptotic formula for the number of solutions of a certain congruence modulo $m$ with four variables, where the variables belong to special sets of residue classes modulo $m.$ This formula are applied…

The metric Mahler measure was first studied by Dubickas and Smyth in 2001 as a means of phrasing Lehmer's conjecture in topological language. More recent work of the author examined a parametrized family of generalized metric Mahler…

For every monic polynomial $f \in \mathbb{Z}[X]$ with $\operatorname{deg}(f) \geq 1$, let $\mathcal{L}(f)$ be the set of all linear recurrences with values in $\mathbb{Z}$ and characteristic polynomial $f$, and let \begin{equation*}…

Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. However, there does not exist any algorithm that…

We consider a certain mixed polynomial which is an extended Lens equation $L_{n,m}=\bar z^m-p(z)/q(z)$ with $\text{degree}\, q=n$, $\text{degree}\, p<n$ whose numerator is a mixed polynomial of degree $(n+m;n,m)$. Then we consider its…

We consider continued fractions \frac{-a_1}{1-\frac{a_2}{1-\frac{a_3}{1-...}}} \label{fr} with real coefficients $a_i$ converging to a limit $a$. S.Ramanujan had stated the theorem (see [ABJL], p.38) saying that if $a\neq\frac14$, then the…

We define an $f$-restricted partition $p_f(n,k)$ of fixed length $k$ given by the bivariate generating series \begin{align*} Q_f(z,u) \coloneqq 1+\sum_{n=1}^{\infty}\sum_{k=1}^{\infty} p_f(n,k) u^kz^n…

Let $\frak E$ denote be the ring of Eisenstein integers. Let $z\in \mathbb C$ and $p_n,q_n \in \frak E$ be such that $\{p_n/q_n\}$ is the sequence of convergents corresponding to the continued fraction expansion of $z$ with respect to the…

In this work, we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin-Iosifescu…

By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial…

This paper is concerned with the study of the fractional finite sums theory. We present the classes of functions for which it is possible to characterize the constant related to the derivative of fractional sums (denominated by essence of a…

We explore Mahler numbers originating from functions $f(z)$ that satisfy the functional equation $f(z) = (A(z)f(z^d) + C(z))/B(z)$. A procedure to compute the irrationality exponents of such numbers is developed using continued fractions…

A classical result of Khinchin says that for almost all real numbers $\alpha$, the geometric mean of the first $n$ digits $a_i(\alpha)$ in the continued fraction expansion of $\alpha$ converges to a number $K = 2.6854520\ldots$ (Khinchin's…

In this paper we prove the following renewal-type limit theorem. Given an irrational $\alpha$ in (0,1) and R>0, let $q_{n_R}$ be the first denominator of the convergents of $\alpha$ which exceeds R. The main result in the paper is that the…

In this paper we consider error sums of the form \[\sum_{m=0}^{\infty} \varepsilon_m\Big( \,b_m\alpha - \frac{a_m}{c_m}\,\Big) \,,\] where $\alpha$ is a real number, $a_m$, $b_m$, $c_m$ are integers, and $\varepsilon_m=1$ or $\varepsilon_m…

Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from…

The purpose of this article is two-folds. Firstly, we establish two sufficient conditions under which the sequence $\{f(n)\pmod{m}: n\geq1\}$ is non-periodic, where $f(n)$ is an arithmetic function. As consequences, we deduce that the…