Related papers: Continued fraction expansions for complex numbers …
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 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…
This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the…
We construct continued fraction expansions for several families of the Laurent series in $\mathbb{Q}[[t^{-1}]]$. To the best of the author's knowledge, this is the first result of this kind since Gauss derived the continued fraction…
For uniformly chosen random $\alpha \in [0,1]$, it is known the probability the $n^{\rm th}$ digit of the continued-fraction expansion, $[\alpha]_n$ converges to the Gauss-Kuzmin distribution $\mathbb{P}([\alpha]_n = k) \approx \log_2 (1 +…
Inspired by several alternative definitions of continued fraction expansions for elements in $\mathbb Q_p$, we study $p$-adically convergent periodic continued fractions with partial quotients in $\mathbb Z[1/p]$. To this end, following a…
We develop a continued fraction algorithm in finite extensions of $\Q_p$ generalising certain algorithms in $\Q_p$, and prove the finiteness property for certain small degree extensions. We also discuss the metrical properties of the…
We establish the strong unique continuation property of fractional orders of linear elliptic equations with Lipschitz coefficients by establishing monotonicity of some Almgren-type frequency functional via an extension procedure.
The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization…
The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate…
We address the problem of determining the Hausdorff dimension of sets consisting of complex irrationals whose complex continued fraction digits satisfy prescribed restrictions and growth conditions. For the Hurwitz continued fraction, we…
The classical Dedekind sums $s(d, c)$ can be represented as sums over the partial quotients of the continued fraction expansion of the rational $\frac{d}{c}$. Hardy sums, the analog integer-valued sums arising in the transformation of the…
We employ some results about continued fraction expansions of Herglotz-Nevanlinna functions to characterize the spectral data of generalized indefinite strings of Stieltjes type. In particular, this solves the corresponding inverse spectral…
We study the topological, dynamical, and descriptive set theoretic properties of Hurwitz continued fractions. Hurwitz continued fractions associate an infinite sequence of Gaussian integers to every complex number which is not a Gaussian…
Let $K$ be a number field. We show that, up to allowing a finite set of denominators in the partial quotients, it is possible to define algorithms for $\mathfrak P$-adic continued fractions satisfying the finiteness property on $K$ for…
We give an elementary geometric proof using Ford circles that the convergents of the continued fraction expansion of a real number $\alpha$ coincide with the rationals that are best approximations of the second kind of $\alpha$.
In this paper we answer certain questions posed by V.I. Arnold, namely, we study periods of continued fractions for solutions of quadratic equations in the form $x^2+p x=q$ with integer $p$ and $q$, $p^2+q^2\le R^2$. Our results concern the…
Recently in joint work with E. Sert, we proved sharp boundedness results on discrete fractional integral operators along binary quadratic forms. Present work vastly enhances the scope of those results by extending boundedness to bivariate…
We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest…
Let $[a_1(x),a_2(x),\ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0, 1)$. The study of the growth rate of the product of consecutive partial quotients $a_n(x)a_{n+1}(x)$ is associated with the…