Pietro Corvaja, Umberto Zannier
Generalizing a result of Pourchet, we prove that, if $α,β$ are power sums satisfying suitable conditions, the length of the continued fraction of the ratio $α(n)/β(n)$ tends to infinity with $n$.
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We state and prove three general formulas allowing to transform formal finite sums into formal continued fractions and apply them to generalize certain expansions in continued fractions given by Hone and Varona.
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 …
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
The present paper proves that if for a power sum $α$ over $\ZZ$ the length of the period of the continued fraction for $\sqrt{α(n)}$ is constant for infinitely many even (resp. odd) $n$, then $\sqrt{α(n)}$ admits a functional continued fraction expansion for all even (resp. odd) $n$, except finitely many; in particular, for such $n$, the partial quotients can be expressed by power sums of the same kind.
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].
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 $n\to\infty$.
On Periodicity of Continued fractions with Partial Quotients in Quadratic Number Fields
针对部分商取自实二次域整数环的连分数，本文研究其最终周期性与收敛性，给出实四次无理数对应周期展开的充要条件及构造算法。
Sums of nearest integer continued fractions with bounded digits: $\textrm{NICF}_5 + \textrm{NICF}_5 = {\mathbb R}$
本研究调整用于正则连分数的康托尔集方法，证明所有实数都可表示为两个部分商不超过5的最近整数连分数对应的实数之和，且该界5无法改进为4。
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.
On continued fraction partial quotients of square roots of primes
本文证明素数平方根连分数中特定部分商出现次数的有限性，给出周期长度可被4整除时部分商1的存在性，并建立周期长度上界及多项式分解，回答Miska和Ulas的若干问题。
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.
Hausdorff dimension of sets of continued fractions with bounded odd and even order partial quotients
研究奇数阶部分商为1的连分数集合的大小，证明两个此类集合的和与积均包含非空区间，并确定奇数阶部分商为1且偶数阶部分商以指数、超指数及一般正函数速率增长时集合的Hausdorff维数。
In the present work, we investigate real numbers whose sequence of partial quotients enjoys some combinatorial properties involving the notion of palindrome. We provide three new transendence criteria, that apply to a broad class of continued fraction expansions, including expansions with unbounded partial quotients. Their proofs heavily depend on the Schmidt Subspace Theorem.
Metric properties of continued fractions with large prime partial quotients
研究连分数展开中存在至少两个同时大且素数的部分商的数集，建立其勒贝格测度的零一定律并确定其Hausdorff维度。
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…
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.
In this paper, we study the combinatorial sum $$\sum_{k\equiv r(\mbox{mod }m)}{n\choose k}a^k.$$ By studying this sum, we obtain new congruences for Lucas quotients of two infinite families of Lucas sequences. Only for three Lucas sequences, there are such known results.
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.
The problem of representing a given positive integer as a sum of four squares of integers has been widely concerned for a long time, and for a given positive odd $n$ one can find a representation by doing arithmetic in a maximal order of quaternion algebra once a pair of (positive) integers $x,y$ with $x^2+y^2\equiv-1\mod n$ is given. In this paper, we introduce a new method to find a representation of odd integer $w$ given $x,y$ satisfying the above requirement. This method can avoid the complicated non-commutative structure in quaternion algebra, which is similar to the one we use to obtain …
We estimate the sum of products or quotients of $L$-functions, where the sum is taken over all quadratic extensions of given genus over a fixed global function field. Our estimate for the sum of the quotient of two $L$-functions is analogous to a result of Schmidt where he estimates the sum of the quotient of two $L$-series, where the sum is over quadratic extensions of $\bq$ with absolute value of the discriminant less than a given bound.
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…
In [16], we obtained some congruences for Lucas quotients of two infinite families of Lucas sequences by studying the combinatorial sum $$\sum_{k\equiv r(\mbox{mod}m)}{n\choose k}a^k.$$ In this paper, we show that the sum can be expressed in terms of some recurrent sequences with orders not exceeding $φ{(m)}$ and give some new congruences.
We ask, for which $n$ does there exists a $k$, $1 \leq k < n$ and $(k,n)=1$, so that $k/n$ has a continued fraction whose partial quotients are bounded in average by a constant $B$? This question is intimately connected with several other well-known problems, and we provide a lower bound in the case of B=2.
Sums of Laurent series with bounded partial quotients
本文研究洛朗级数中具有有界部分商的实数表示问题，提出新的结果以扩展Hall和Cusick的经典结论。
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)…
In this paper we study certain real functions defined in a very simple way by Zagier as sums of infinite powers of quadratic polynomials with integer coefficients. These functions give the even parts of the period polynomials of the modular forms which are the coefficients in Fourier expansion of the kernel function for Shimura-Shintani correspondence. We prove two conjectures of Zagier showing that the sums converge exponentially. We also prove unexpected results on the representation of these functions as sums over simple or reduced quadratic forms and the positive or negative continued frac…
For a prime $p$ and an integer $u$ with $\gcd(u,p)=1$, we define Fermat quotients by the conditions $$ q_p(u) \equiv \frac{u^{p-1} -1}{p} \pmod p, \qquad 0 \le q_p(u) \le p-1. $$ D. R. Heath-Brown has given a bound of exponential sums with $N$ consecutive Fermat quotients that is nontrivial for $N\ge p^{1/2+ε}$ for any fixed $ε>0$. We use a recent idea of M. Z. Garaev together with a form of the large sieve inequality due to S. Baier and L. Zhao, to show that on average over $p$ one can obtain a nontrivial estimate for much shorter sums starting with $N\ge p^ε$. We also obtain lower bounds on …
Metrical properties for the large partial quotients with product forms in continued fractions
本文研究了连分数中前n项内至少两个具有乘积形式的大偏商的分布，完全确定了相关集合在Lebesgue测度和Hausdorff维度下的大小，丰富了度量丢番图逼近理论。
It is widely believed that the continued fraction expansion of every irrational algebraic number $α$ either is eventually periodic (and we know that this is the case if and only if $α$ is a quadratic irrational), or it contains arbitrarily large partial quotients. Apparently, this question was first considered by Khintchine. A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The main purpose of the present work is to present new families of transcendental continued fractions with bounded partial quotients. Our results are de…
Representations of integers as quotients of sums of distinct powers of three
本文研究哪些整数可表示为不同三的幂之和的商，提出必要且几乎充分的条件和生成所有表示的算法，并基于0,1-多项式对表示进行分类，完整描述了不超过364的整数及若干无限族。
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…
Anomalous Partial Quotients in the Continued Fraction of $\sqrt{ζ(3)-S_N}$
研究ζ(3)-S_N的连分数展开，通过Euler-Maclaurin展开及Pell恒等式确定特定指标下部分商规律，给出前五个部分商公式，揭示异常增长现象。
Eta-Quotient Representations for a Three Parameter Family of Modular Functions Associated with the Rogers-Ramanujan Continued Fraction
本文构造了与Rogers-Ramanujan连分数相关的三参数模函数族，推导其递归eta商表示，并应用于剖分公式，特别是用于获得受限奇数差分拆的5-剖分公式。
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.
The continued fraction expansion of an irrational number $α$ is eventually periodic if and only if $α$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three. Because of some numerical evidence and a belief that these numbers behave like most numbers in this respect, it is often conjectured that their partial quotients form an unbounded sequence. More modestly, we may expect that if the sequence of partial quotients of an irrational number $α$ is, in some sense, "simple", then $α$ is either quad…
Continued Fractions and Irrationality Measures for Chowla--Selberg Gamma Quotients
研究Chowla - Selberg伽马商，通过给出39个快速收敛连分数，推导出其中20个的无理测度，如\(\operatorname{CS}(-3)\)等，这是关于伽马商的首个经证明且合理的无理测度。
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…
Continued fractions with large prime partial quotients
本文研究部分商为大素数的连分数实数集合，通过证明几乎素数zeta函数尾端的新渐近结果，确定了相应集合的勒贝格测度与豪斯多夫维数，推广并涵盖了多个已有经典与近期定理。
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…
In this note we consider a question of Ono, concerning which spaces of classical modular forms can be generated by sums of $η$-quotients. We give some new examples of spaces of modular forms which can be generated as sums of $η$-quotients, and show that we can write all modular forms of level $Γ_0(N)$ as rational functions of $η$-products.
Arithmetic sums and products of infinite multiple zeta-star values
本文研究了受限索引的无穷多重ζ星值的算术和与积，并受连分数理论和康托集启发，提出关于特定索引的ζ星值代数点及算术运算的猜想。
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.
Hausdorff dimension of sets with restricted, slowly growing partial quotients in semi-regular continued fractions
本研究确定了半正则连分数中部分商受限且缓慢增长的无理数集合的豪斯多夫维数，推广了正则连分数的相关结果，通过构造非自治迭代函数系统并应用Bowen公式证明。
In this paper we consider error sums of the form \[\sum_{m=0}^{\infty} \varepsilon_m\Big( \,b_mα- \frac{a_m}{c_m}\,\Big) \,,\] where $α$ is a real number, $a_m$, $b_m$, $c_m$ are integers, and $\varepsilon_m=1$ or $\varepsilon_m ={(-1)}^m$. In particular, we investigate such sums for \[α\in \big\{ π, e,e^{1/2},e^{1/3},\dots, \log (1+t), ζ(2), ζ(3) \big\} \] and exhibit some connections between rational coefficients occurring in error sums for Apéry's continued fraction for $ζ(2)$ and well-known integer sequences. The concept of the paper generalizes the theory of ordinary error sums, which are…
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.
We generalise remarks of Euler and of Perron by explaining how to detail all quadratic irrational integers for which the symmetric part of the period of their continued fraction expansion commences with prescribed partial quotients. The function field case is particularly striking.
The arithmetic of continued fractions in the field of $p$-adic numbers
本文研究了p进数域中连分数的算术，提出了一种计算莫比乌斯变换和双线性分数变换的完整方法，并证明了某些情况下部分商无法通过初始连分数恢复，但此类元素集合在p进数域上具有零测度。
The purpose of this paper is to construct p-adic analytically continued function which interpolates q-Euler numbers at negative integer Finally, we give an explicit p-adic expansion as a power series in n.
Eta quotients on $Γ_0(6)$ yield evaluations of sunrise integrals at 2, 3, 4 and 6 loops. At 2 and 3 loops, they provide modular parametrizations of inhomogeneous differential equations whose solutions are readily obtained by expanding in the nome $q$. Atkin-Lehner transformations that permute cusps ensure fast convergence for all external momenta. At 4 and 6 loops, on-shell integrals are periods of modular forms of weights 4 and 6 given by Eichler integrals of eta quotients. Weakly holomorphic eta quotients determine quasi-periods. A Rademacher sum formula is given for Fourier coefficients of …
Metric Theory for Continued Fractions with Multiple Large Partial Quotients
本文研究连分数展开前n项中至少含有r个大偏商的实数集合，确定其Lebesgue测度和Hausdorff维数，解决了多个大偏商情形下经典极限定理失效的问题。
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.
Multifractal analysis of maximal product of consecutive partial quotients in continued fractions
本文从多重分形分析角度研究连分数前n项连续部分商最大乘积的增长速率，确定了对应水平集的Hausdorff维数，发现ρ=1/2时维数存在跳跃，并构造不连续函数填补了该维数间隙。
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.
Bounded continued fractions and quadratic orders in $\Q(\sqrt{127})$
本文研究Q(√127)中部分商≤4的有界连分数的周期与乘子序，证明相关精确公式，结合Hensley定理得到二次域允许无穷多此类连分数的数量下界。
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 $4$.
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