Let $[a_1(x),a_2(x),a_3(x),\cdots]$ be the continued fraction expansion of $x\in (0,1)$. This paper is concerned with certain sets of continued fractions with non-decreasing partial quotients. As a main result, we obtain the Hausdorff dimension of the set \[\left\{x\in(0,1): a_1(x)\leq a_2(x)\leq \cdots,\ \limsup\limits_{n\to\infty}\frac{\log a_n(x)}{ψ(n)}=1\right\}\] for any $ψ:\mathbb{N}\rightarrow\mathbb{R}^+$ satisfying $ψ(n)\to\infty$ as $n\to\infty$.
Consider the representation of a rational number as a continued fraction, associated with "odd" Euclidean algorithm. In this paper we prove certain properties for the limit distribution function for sequences of rationals with bounded sum of partial quotients
In 1986, some examples of algebraic, and nonquadratic, power series over a finite prime field, having a continued fraction expansion with partial quotients all of degree one, were discovered by W. Mills and D. Robbins. In this note we show how these few examples are included in a very large family of continued fractions for certain algebraic power series over an arbitrary finite field of odd characteristic.
Let $h$: $x\mapsto \frac{ax+b}{cx+d} $ be the nondegenerate Möbius transformation with integer entries. We get a bound of the continued fraction of $h(x)$ by the upper and lower bound of continued fraction of $x$, which extends a result of Stambul \cite{S2}.
In this note, we describe a family of particular algebraic, and nonquadratic, power series over an arbitrary finite field of characteristic 2, having a continued fraction expansion with all partial quotients of degree one. The main purpose being to point out a common origin with another analogue family in odd characteristic, which has been studied in [LY1].
We present and develop different approaches to study the asymptotic behavior of the distribution functions in the odd continued fractions case. Firstly, by considering the transition operator of the Markov chain associated with these expansions on a certain Banach space of complex-valued functions of bounded variation we make a brief survey of the solution in the Gauss-Kuzmin-type problem. Secondly, we use the method of Szüsz to obtain a similar asymptotic result and to give a good estimate of the convergence rate involved.
Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow Kuzmin's probability law. Results are given for sequences of partial quotients of $\sqrt[3]{m}$ and $\sqrt[3]{m^2}$ with $m$ noncube. A big partial quotient in one sequence finds a connection in the other.
We prove results concerning the joint limiting distribution of the renewal time of denominators and consecutive digits of random irrational numbers in the case of continued fractions with even partial quotients, with odd partial quotients, and for Nakada's α-expansions.
In this paper we study the three-dimensional analogue of the relation between the irrationality exponent of a real number and the growth of its regular continued fraction partial quotients. As a multidimensional generalisation of continued fractions, we consider Klein polyhedra.
We construct a class of quadratic irrationals having continued fractions of period $n\geq2$ with "small" partial quotients for which certain integer multiples have continued fractions of period $1$, $2$ or $4$ with "large" partial quotients. We then show that numbers in the period of the new continued fraction are simple functions of the numbers in the periods of the original continued fraction. We give generalizations of some of the continued fractions and show that polynomials arising from the generalizations are related to Chebyshev and Fibonacci polynomials. We then show that some of these…
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 logarithms of $θ$-functions under a subgroup of the modular group, have been shown to satisfy many properties which mirror the properties of the classical Dedekind sums. The representation as sums of partial quotients has, however, been missing so far. We define non-classical continued fractions and prove that Hardy sums can be expressed as a sums of partial …
Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow Kuzmin's probability law. Results are given for sequences of partial quotients of positive irrational numbers $ξ$ and $m \over ξ$ with $m$ a natural number. A big partial quotient in one sequence finds a connection in the other.
We consider sets of real numbers in $[0,1)$ with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by $1/2$, are given by a modified variational principle.
We introduce here a general framework for studying continued fraction expansions for complex numbers and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial quotients in a discrete subring of $\mathbb C$ an analogue of the classical Lagrange theorem, characterising quadratic surds as numbers with eventually periodic continued fraction expansions, is proved. Monotonicity and exponential growth are established for the absolute values of the denominators of the convergents for a class of continued fraction algorith…
For a real $x\in(0,1)\setminus\mathbb{Q}$, let $x=[a_1(x),a_2(x),\cdots]$ be its continued fraction expansion. Denote by $T_n(x):= max \{a_k(x): 1\leq k\leq n\}$ the leading partial quotient up to $n$. For any real $α\in(0,\infty), γ\in(0,\infty)$, let $F(γ,α):=\{x\in(0,1)\setminus\mathbb{Q}: \lim_{n\rightarrow\infty}\frac{T_n(x)}{e^{n^γ}}=α\}$. For a set $E\subset (0,1)\setminus\mathbb{Q}$, let $dim_H E$ be its Hausdorff dimension. Recently Lingmin Liao and Michal Rams [LR, Theorem 1.3] show that $dim_H F(γ,α)$ is $1$ if $r\in(0,1/2)$, it is $1/2$ if $r\in(1/2,\infty)$ for any $α\in(0,\infty)…
V.I. Arnold has experimentally established that the limit of the statistics of incomplete quotients of partial continued fractions of quadratic irrationalities coincides with the Gauss--Kuz'min statistics. Below we briefly prove this fact for roots of the equation $r x^2+p x=q$ with fixed $p$ and $r$ ($r>0$), and with random $q$, $q\le R$, $R\to \infty$. In Section 3 we estimate the sum of incomplete quotients of the period. According to the obtained bound, prior to the passage to the limit, incomplete quotients in average are logarithmically small. We also upper estimate the proportion of the…
Consider the representation of a rational number in the form, associated with "centered" Euclidean algorithm. We prove a new formula for the limit distribution function for sequences of rationals with bounded sum of partial quotients.
It is possible to define a continued fraction expansion of elements in a function field of a curve by expanding as a Laurent series in a local parameter. Considering the square root of a polynomial $\sqrt{D(t)}$ leads to an interesting theory related to polynomial Pell equations. Unlike the classical Pell equation, the corresponding polynomial equation is not always solvable and its solvability is related to arithmetic conditions on the Jacobian (or generalized Jacobian) of the curve defined by $y^2=D(t)$. In this setting, it has been shown by Zannier in \cite{zannier} that the sequence of the…
The classical Khintchine and Jarník theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real numbers for which the partial quotients in their continued fraction expansions grows with a certain rate. Recently it was observed that the growth of product of pairs of consecutive partial quotients in the continued fraction expansion of a real number is associated with improvements to Dirichlet's theorem. In this paper we consider the products of several consecut…
We introduced a new continued fraction expansions in our previous paper. For these expansions, we show formulae of probability about incomplete quotients. Furthermore, we prove the existence of invariant measures with respect to the continued fraction transformations associated with expansions.
We establish a new transcendence criterion of $p$-adic continued fractions which are called Ruban continued fractions. By this result, we give explicit transcendental Ruban continued fractions with bounded $p$-adic absolute value of partial quotients. This is $p$-adic analogy of Baker's result. We also prove that $p$-adic analogy of Lagrange Theorem for Ruban continued fractions is not true.
A classical theorem in continued fractions due to Serret shows that for any two irrational numbers x and y related by a transformation $γ$ in PGL(2,Z) there exist s and t for which the complete quotients x_s and y_t coincide. In this paper we give an upper bound in terms of $γ$ for the smallest indices s and t.
We study a family of generalized continued fractions, which are defined by a pair of substitution sequences in a finite alphabet. We prove that they are stammering sequences, in the sense of Adamczewski and Bugeaud. We also prove that this family consists of transcendental numbers which are not Liouvillian. We explore the partial quotients of their regular continued fraction expansions, arriving at no conclusion concerning their boundedness.
A converse method to the Construction of Salem (1945) of convergent families of Salem numbers is investigated in terms of an association between Salem polynomials and Hurwitz quotients via expansive polynomials of small Mahler measure. This association makes use of Bertin-Boyd's Theorem A (1995) of interlacing of conjugates on the unit circle; in this context, a Salem number $β$ is produced and coded by an m-tuple of positive rational numbers characterizing the (SITZ) Stieltjes continued fraction of the corresponding Hurwitz quotient (alternant). The subset of Stieltjes continued fractions ove…
Let $a, b$ be distinct, non-constant polynomials in ${\mathbb F}_2 [z]$. Let $ξ_{a, b}$ be the power series in ${\mathbb F}_2((z^{-1}))$ whose sequence of partial quotients is the Thue-Morse sequence over $\{a, b\}$. We establish that $ξ_{a,b}$ is algebraic of degree $4$.
We look at the rate of growth of the partial quotients of the infinite continued fraction expansion of an irrational number relative to the rate of approximation of the number by its convergents. In non-generic cases the Hausdorff dimension of some exceptional sets is computed.
We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_1,a_2,\ldots,a_n]$, we associate a snake graph $\mathcal{G}[a_1,a_2,\ldots,a_n]$ such that the continued fraction is the quotient of the number of perfect matchings of $\mathcal{G}[a_1,a_2,\ldots,a_n]$ and $\mathcal{G}[a_2,\ldots,a_n]$. We also show that snake graphs are in bijection with continued fractions. We then apply this conn…
We establish new combinatorial transcendence criteria for continued fraction expansions. Let $α= [0; a_1, a_2,...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients $(a_{\ell})_{\ell \ge 1}$ of $α$ cannot be generated by a finite automaton, and that the complexity function of $(a_{\ell})_{\ell \ge 1}$ cannot increase too slowly.
Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to generalize the classical continued fractions. In this paper, we propose an introductive fundamental study about MCFs in the field of the $p$--adic numbers $\mathbb Q_p$. First, we introduce them from a formal point of view, i.e., without considering a specific algorithm that produces the partial quotients of a MCF, and we perform a general study about their convergence in $\mathbb Q_p$. In particular, we derive some conditions about their convergence and we prove that convergent MCFs always strongly co…
We show that for a fixed integer $n \neq \pm2$, the congruence $x^2 + nx \pm 1 \equiv 0 \pmodα$ has the solution $β$ with $0 < β< α$ if and only if $α/β$ has a continued fraction expansion with sequence of quotients having one of a finite number of possible asymmetry types. This generalizes the old theorem that a rational number $α/β> 1$ in lowest terms has a symmetric continued fraction precisely when $β^2 \equiv \pm 1 \pmodα$.
For a complex polynomial $D(t)$ of even degree, one may define the continued fraction of $\sqrt{D(t)}$. This was found relevant already by Abel in 1826, and later by Chebyshev, concerning integration of (hyperelliptic) differentials; they realized that, contrary to the classical case of square roots of positive integers treated by Lagrange and Galois, we do not always have pre-periodicity of the partial quotients. In this paper we shall prove that, however, a correct analogue of Lagrange's theorem still exists in full generality: pre-periodicity of the {\it degrees} of the partial quotients al…
Let $x \in [0,1)$ be a real number and denote its continued fraction expansion by $[a_1(x),a_2(x), a_3(x),\cdots]$. The convergence exponent of these partial quotients is defined as \[ τ(x):= \inf\left\{s \geq 0: \sum_{n \geq 1} a^{-s}_n(x)<\infty\right\}. \] In this paper, we investigate some fundamental properties and multifractal analysis of the exponent $τ(x)$.
We apply a technique used in $[$Tsukerman, Equality of Dedekind sums mod $\mathbb Z$, $2\mathbb Z$ and $4\mathbb Z$, arXiv:1408.3225] combined with the Barkan-Hickerson-Knuth-formula in order to obtain congruences mod $4$ for the alternating sum of the partial quotients of the continued fraction expansion of $a/b$, where $0<a<b$ are integers.
Large and moderate deviation principles are proved for Engel continued fractions, a new type of continued fraction expansion with non-decreasing partial quotients in number theory.
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 provides a periodic multidimensional continued fraction when algebraic irrationalities are given as inputs. In this paper, we provide a characterization for periodicity of Jacobi--Perron algorithm by means of linear recurrence sequences. In particular, we prove that partial quotients of a multidimensional continued fraction are periodic if and only if numerato…
Irrationality Exponents and Partial Quotient Growth in Continued Fractions
School of Mathematics, South China University of Technology（华南理工大学数学学院）；School of Mathematics and Systems Science, & Center for Mathematical Sciences, Wuhan University of Science and Technology（武汉科技大学数学与系统科学学院及数学科学中心）
By replacing the letters to polynomials in F_2[t], an infinite word, over a finite alphabet, can be seen as the sequence of partial quotients of a continued fraction in F_2((1/t)). Here is described a family of such infinite words, corresponding to continued fractions which are algebraic over F_2(t). This family includes a classical example already studied in different previous works by Y. Hu, G-N. Han, Y. Bugeaud and the author of this note.
