<Emphasis Type="Italic">P</Emphasis>-adic continued fractions

We study Schneider’s p-adic continued fraction algorithms. For p=2, we give a combinatorial characterization of rational numbers that have terminating expansions. For arbitrary p, we give data showing that rationals with terminating expansions are relatively rare. Finally, we prove an analogue of Khinchin’s theorem.     

Continued fractions for hyperquadratic power series over a finite field

An irrational power series over a finite field of characteristic p is called hyperquadratic if it satisfies an algebraic equation of the form x=(Ax^r+B)/(Cx^r+D), where r is a power of p and the coefficients belong to . These algebraic power series are analogues of quadratic real numbers. This analogy makes their continued fraction expansions specific as in the classical case, but more sophisticated. Here we present a general result on the way some of these expansions are generated. We apply it to describe several families of expansions having a regular pattern.     

Simple continued fraction solutions for diophantine equations

We present elementary necessary and sufficient conditions for the solvability of the Diophantine equation x² − Dy² = n for any n and any nonsquare integer D > 0, using only simple continued fraction expansions. This includes a simple device for finding fundamental solutions of such equations.     

Red numbers and quadratic field

A natural number is said red if the period of the continued fraction of its square root has odd length. For any quadratic field \mathbbQ(?D)\mathbb{Q}(\sqrt{D}), we show how the parity of the periods length of the continued fractions of its irrationalities depends on the redness of their discriminant.     

Diophantine Approximations on Fractals

We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0, 1]², possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no irrational numbers x in the unit interval such that the continued fraction expansions of {nx mod 1}_{n ? \mathbb N}{\{nx\,{\rm mod}\,1\}_{n \in {\mathbb N}}} are uniformly eventually bounded.     

Comparison of various generalizations of continued fractions

We use the Euler, Jacobi, Poincaré, and Brun matrix algorithms as well as two new algorithms to evaluate the continued fraction expansions of two vectorsL related to two Davenport cubic formsg ₁ andg ₂. The Klein polyhedra ofg ₁ andg ₂ were calculated in another paper. Here the integer convergentsP _k given by the cited algorithms are considered with respect to the Klein polyhedra. We also study the periods of these expansions. It turns out that only the Jacobi and Bryuno algorithms can be regarded as satisfactory. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 339–348, March, 1997. Translated by V. E. Nazaikinskii     

On the period of the continued fraction expansion of √2 + 1

We prove a general result which implies that the period of the continued fraction expansion of√2²ⁿ⁺¹ + 1 tends to infinity when n tends to infinity.     

A polynomial algorithm for maximum weighted vertex packings on graphs without long odd cycles

The vertex packing problem for a given graph is to find a maximum number of vertices no two of which are joined by an edge. The weighted version of this problem is to find a vertex packingP such that the sum of the individual vertex weights is maximum. LetG be the family of graphs whose longest odd cycle is of length not greater than 2K + 1, whereK is any non-negative integer independent of the number (denoted byn) of vertices in the graph. We present an O(n ^2K+1) algorithm for the maximum weighted vertex packing problem for graphs inG 1. A by-product of this algorithm is an algorithm for piecing together maximum weighted packings on blocks to find maximum weighted packings on graphs that contain more than one block. We also give an O(n ^2K+3) algorithm for testing membership inG This work was supported by NSF grant ENG75-00568 to Cornell University. Some of the results of this paper have appeared in Hsu's unpublished Ph.D. dissertation [9].     

A simple proof of a classical identity

A few well-known infinite partial fraction expansions of some trigonometric functions are derived and Euler's elegant identity ∑^∞_{k = 1}1/k² = π²/6 is deduced as a corollary.     

On the length of the homotopic Brownian word in the thrice punctured sphere

We consider the word associated to the homotopic class of the Brownian path (properly closed) in the thrice punctured sphere. We prove that its length has almost surely the same behaviour as a totally asymmetric Cauchy process on the line. More precisely, the liminf has the same normalization in t log(t) and the limsup can be described by the same integral test. They are the Brownian motion counterparts of some Lévy and Khintchine results on continued fraction expansions. Received: 17 December 1996 / Revised version: 23 February 1998     

About statistics of periods of continued fractions of quadratic irrationalities

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+px=q with integer p and q, p ²+q ²≤R ². We prove a weak variant of Arnold conjectures about the Gauss–Kuzmin statistics with R→∞.     

Continued fractions for some alternating series

We discuss certain simple continued fractions that exhibit a type of “self-similar” structure: their partial quotients are formed by perturbing and shifting the denominators of their convergents. We prove that all such continued fractions represent transcendental numbers. As an application, we prove that Cahen's constant $$C = \sum\limits_{i \geqslant 0} {\frac{{( - 1)^i }}{{S_i - 1}}}$$ is transcendental. Here (S _n ) isSylvester's sequence defined byS ₀=2 andS _n+1 =S _n ² ?S _n +1 forn≥0. We also explicitly compute the continued fraction for the numberC; its partial quotients grow doubly exponentially and they are all squares.     

On the Hausdorff dimension of fractals given by certain expansions of real numbers

We transfer classical results on the Hausdorff dimension of b-adic and continued fraction expansions of real numbers to another expansion.     

On the moments of elements of continued fractions for some rational numbers

Let p be a prime and let 1 ≤ a ≤ p − 1. In the paper, an asymptotics for the sum over a of the moments of order α (0 < α < 1) of the sequence of elements of the expansion of a/p into a continued fraction is obtained. As a corollary, an upper bound for the number of those a whose expansions contain at least one element larger than log ^λ p (λ > 1) is derived. Note that in the case considered, the set of elements has no limiting distribution as p → ∞, which is in contrast with the case of rational fractions b/c, where (b, c) = 1 and b² + c² ≤ R² (R → ∞). Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 13–22.     

Transplanting maximal inequalities between Laguerre and Hankel multipliers

We supplement our previous paper [9] by adding a theorem that transplantsL ^p -norm maximal inequalities for Laguerre multipliers. As an immediate consequence we obtain negative results concerningL ^p -estimates of partial sum maximal operators for Laguerre expansions.Research supported in part by KBN grant No. 2 PO3A 030 09.     

Asymptotic Distribution of Quadratic Forms and Applications

We consider the quadratic formsQ X _j X _k+ (X _j ² -E X _j ² )where X _j are i.i.d. random variables with finite sixth moment. For a large class of matrices (a _jk ) the distribution of Q can be approximated by the distribution of a second order polynomial in Gaussian random variables. We provide optimal bounds for the Kolmogorov distance between these distributions, extending previous results for matrices with zero diagonals to the general case. Furthermore, applications to quadratic forms of ARMA-processes, goodness-of-fit as well as spacing statistics are included.     

On cardinalities of row spaces of Boolean matrices

We prove that there are exactlyn numbers greater than 2 ⁿ⁻¹ that can serve as the cardinalities of row spaces ofn×n Boolean matrices. The numbers are: 2 ⁿ⁻¹+1,2 ⁿ⁻¹+2,2 ⁿ⁻¹+4, ..., 2 ⁿ⁻¹+2 ⁿ⁻², 2 ⁿ . Two consequences follow. The first is that the height of the partial order ofD-classes in the semigroup ofn×n Boolean matrices is at most 2 ⁿ⁻¹+n−1. The second is that the numbers listed above are precisely the numbers greater than 2 ⁿ⁻¹ that can serve as the cardinalities of topologies on a finite setX withn elements.     

Nota sulle frazioni continuep-adiche

Summary In this note it is proved that for p-adic continued fractions a result analogous to Galois theorem for ordinary continued fractions holds. Moreover some results concerning the p- adic continued fraction expansions of the square roots of the integers are obtained.     

On the amicability of orthogonal designs

Although it is known that the maximum number of variables in two amicable orthogonal designs of order 2ⁿp, where p is an odd integer, never exceeds 2n+2, not much is known about the existence of amicable orthogonal designs lacking zero entries that have 2n+2 variables in total. In this paper we develop two methods to construct amicable orthogonal designs of order 2ⁿp where p odd, with no zero entries and with the total number of variables equal or nearly equal to 2n+2. In doing so, we make a surprising connection between the two concepts of amicable sets of matrices and an amicable pair of matrices. With the recent discovery of a link between the theory of amicable orthogonal designs and space‐time codes, this paper may have applications in space‐time codes. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 240‐252, 2009