Dyadic fractions with small partial quotients

It is proved that ifm is a power of 2, then there exists an odd integera with 1a<m such that all partial quotients in the continued fraction expansion ofa/m are bounded by 3. The upper bound 3 is best possible. Similar results can be shown for powers of other small numbers.     

Polynomials with Odd Orthogonal Multiplicity

Let theorthogonal multiplicityof a monic polynomialgover a field be the number of polynomialsfover , coprime togand of degree less than that ofg, such that all the partial quotients of the continued fraction expansion off/gare of degree 1. Polynomials with positive orthogonal multiplicity arise in stream cipher theory, part of cryptography, as the minimal polynomials of the initial segments of sequences which have perfect linear complexity profiles. This paper focuses on polynomials which have odd orthogonal multiplicity; such polynomials are characterized and a lower bound on their orthogonal multiplicity is given. A special case of a conjecture on rational functions over the finite field of two elements with partial quotients of degree 1 or 2 in their continued fraction expansion is also proved.     

The Borel—Bernstein Theorem for multidimensional continued fractions

A central result in the metric theory of continued fractions, the Borel—Bernstein Theorem gives statistical information on the rate of increase of the partial quotients. We introduce a geometrical interpretation of the continued fraction algorithm; then, using this set-up, we generalize it to higher dimensions. In this manner, we can define known multidimensional algorithms such as Jacobi—Perron, Poincaré, Brun, Rauzy induction process for interval exchange transformations, etc. For the standard continued fractions, partial quotients become return times in the geometrical approach. The same definition holds for the multidimensional case. We prove that the Borel—Bernstein Theorem holds for recurrent multidimensional continued fraction algorithms. Supported by a grant from the CNP _q -Brazil, 301456/80, and FINEP/CNP _q /MCT 41.96.0923.00 (PRONEX).     

Large and Moderate Deviation Principles for Engel Continued Fractions

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.     

On a New Continued Fraction Expansion with Non-Decreasing Partial Quotients

We investigate metric properties of the digits occurring in a new continued fraction expansion with non-decreasing partial quotients, the so-called Engel continued fraction (ECF) expansion.     

On a New Continued Fraction Expansion with Non-Decreasing Partial Quotients

We investigate metric properties of the digits occurring in a new continued fraction expansion with non-decreasing partial quotients, the so-called Engel continued fraction (ECF) expansion.     

Sur les grands quotients partiels du développement en fraction continue

In this paper we obtain some metrics results about large partial quotients in the continued fraction expansion.     

Congruence Restricted Modular Forms

Given f(z), a modular form on a congruence subgroup (of the full modular group), we construct the function f(z;r,t) by summing over the terms of the Fourier expansion of f(z) with index congruent to r modulo t. In this paper, we determine a condition on the multiplier system of f(z) which guarantees that f(z;r,t) is itself a modular form on a (smaller) congruence subgroup.2000 Mathematics Subject Classification: Primary—11F11; Secondary—11F30     

Continued fractions for certain algebraic power series

Let F be an arbitrary field and let K = F((x⁻¹)) be the field of formal Laurent series in x⁻¹ over F. The usual theory of continued fractions carries over to K, with the polynomials in x playing the role of the integers. We study the continued fraction expansions of elements of K which are algebraic over F(x), the field of rational functions of x.We give the first explicit expansions of algebraic elements of degree greater than 2 for which the degrees of the partial quotients are bounded. In particular we give explicitly the continued fraction expansion for the solution f in K of the cubic equation xf³ + f + x = 0 when F = GF(2). This cubic was studied by Baum and Sweet. We give examples, for every field F of characteristic greater than 2, of algebraic elements of degree greater than 2 whose partial quotients are all linear, and we give these expansions explicitly. These are the first known examples with partial quotients of bounded degree when F has characteristic greater than 2.     

Engel连分数展式中的若干度量性质

摘要：研究了Engel连分数展式的度量性质．与普通连分数一样，证明了部分商的增长性满足0-1率．通过构造一族恰当的集合，得到了部分商增长速度的上下极限．     

Simple continued fractions and cutting sequences

Abstract

The connection between cutting sequences of a directed geodesic in the tessellated hyperbolic plane ?², the modular group Γ = PSL(2, ?) and the simple continued fractions of an end point w of the geodesic have been established by Series [13]. In this paper we represent the simple continued fractions of w ∈ ? and the “L” and “R” codes of the cutting sequence in terms of modular and extended modular transformations. We will define a T ₀-path on a graph whose vertices are the set of Farey triangles, as the equivalent of the cutting sequence. The relationship between the directed geodesic with end point w on ?, the Farey tessellation and the simple continued fraction expansion of w ∈ ? _∞ then follows easily as a consequence of this redefinition. Finite, infinite and periodic simple continued fractions are subsequently examined in this light.     

Cantor sets determined by partial quotients of continued fractions of Laurent series

In this paper, two types of general sets determined by partial quotients of continued fractions over the field of formal Laurent series with coefficients from a given finite field are studied. The Hausdorff dimensions of and are determined completely, where A_n(x) denotes the partial quotients in the continued fraction expansion (in case of Laurent series) of x and (n) is a positive valued function defined on natural numbers N.     

Heegner Divisors and Nonholomorphic Modular Forms

We consider an embedded modular curve in a locally symmetric space M attached to an orthogonal group of signature (p, 2) and associate to it a nonholomorphic elliptic modular form by integrating a certain theta function over the modular curve. We compute the Fourier expansion and identify the generating series of the (suitably defined) intersection numbers of the Heegner divisors in M with the modular curve as the holomorphic part of the modular form. This recovers and generalizes parts of work of Hirzebruch and Zagier.     

The mean number of steps in the Euclidean algorithm with odd partial quotients

The length of the continued-fraction expansion of a rational number with odd partial quotients is expressed via the Gauss-Kuz’min statistics for the classical continued fraction. This has made it possible to prove asymptotic formulas, similar to those already known for the classical Euclidean algorithm, for the mean length of the Euclidean algorithm with odd partial quotients.     

Differential equations for septic theta functions

We demonstrate that quotients of septic theta functions appearing in Ramanujan’s Notebooks and in Klein’s work satisfy a new coupled system of nonlinear differential equations with symmetric form. This differential system bears a close resemblance to an analogous system for quintic theta functions. The proof extends an elementary technique used by Ramanujan to prove the classical differential system for normalized Eisenstein series on the full modular group. In the course of our work, we show that Klein’s quartic relation induces symmetric representations for low-weight Eisenstein series in terms of weight one modular forms of level seven.     

On two conjectures of P. Chowla and S. Chowla concerning continued fractions

Summary The alternating sum of the partial quotients in the primitive period of a continued fraction expansion of √D is determined mod 2 and mod 3. To Professor BeniaminoSegre on his 70-th birthday. Pervenuto il 7 maggio 1973.     

Structure of <Emphasis Type="Italic">K</Emphasis>-interval exchange transformations: Induction,trajectories, and distance theorems

We define a new induction algorithm for k-interval exchange transformations associated to the “symmetric” permutation i ↦ k − i + 1. Acting as a multi-dimensional continued fraction algorithm, it defines a sequence of generalized partial quotients given by an infinite path in a graph whose vertices, or states, are certain trees we call trees of relations. This induction is self-dual for the duality between the usual Rauzy induction and the da Rocha induction. We use it to describe those words obtained by coding orbits of points under a symmetric interval exchange, in terms of the generalized partial quotients associated with the vector of lengths of the k intervals. As a consequence, we improve a bound of Boshernitzan in a generalization of the three-distances theorem for rotations. However, a variant of our algorithm, applied to a class of interval exchange transformations with a different permutation, shows that the former bound is optimal outside the hyperelliptic class of permutations.     

The explicit formulas and evaluations of Ramanujan's theta-function ψ

We define two quotients of theta-function ψ depending on two positive real parameters. We then show how they are connected with two parameters of Dedekind eta-function, theta-function φ, and the Ramanujan-Weber class invariants. Explicit formulas for determining values of the theta-function ψ are derived, and several examples will be given. In addition, we give some applications of these parameters for the famous Rogers-Ramanujan continued fraction R(q), Ramanujan's cubic continued fraction G(q), and the modular j-invariant.     

Arithmetic properties of the Shimura–Shintani–Waldspurger correspondence

We prove that the theta correspondence for the dual pair , for B an indefinite quaternion algebra over ℚ, acting on modular forms of odd square-free level, preserves rationality and p-integrality in both directions. As a consequence, we deduce the rationality of certain period ratios of modular forms and even p-integrality of these ratios under the assumption that p does not divide a certain L-value. The rationality is applied to give a direct construction of isogenies between new quotients of Jacobians of Shimura curves, completely independent of Faltings’ isogeny theorem. To my parents.     

Theta series and trace operators for Г<Subscript>n</Subscript>[q]

We consider the action of suitable trace operators on non homogeneous theta series that are Siegel modular forms for the principal congruence subgroups of the symplectic group of odd levelq: Г _n [q]. This is used for investigating whether modular forms forГ _n [N], withN|q, which are linear combination of such theta series, can be expressed as combination of theta series that are modular forms with respect toГ _n [N].