A Gauss-Kusmin theorem for optimal continued fractions

A Gauss-Kusmin theorem for the Optimal Continued Fraction (OCF) expansion is obtained. In order to do so, first a Gauss-Kusmin theorem is derived for the natural extension of the ergodic system underlying Hurwitz's Singular Continued Fraction (SCF) (and similarly for the continued fraction to the nearer integer (NICF)). Since the NICF, SCF and OCF are all examples of maximal -expansions, it follows from a result of Kraaikamp that the SCF and OCF are metrically isomorphic. This isomorphism is then used to carry over the results for the SCF to any other maximal -expansion, in particular to the OCF. Along the way, a Heilbronn-theorem is obtained for any maximal -expansion.

     

Pringsheim's theorem for generalized continued fractions

In this paper we present a generalization to generalized continued fractions of Pringsheim's theorem on the convergence of ordinary continued fractions.     

Bestness of the parabola theorem for continued fractions

We show that the simple conditional convergence set

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fixed is maximal in the sense that if b is an arbitrary point in , then P{b} is not conditional convergence set.     

Some local properties of two-dimensional continued fractions

By regarding a two-dimensional continued fraction as a function of its elements and applying recursion relations for its tails, we establish formulas for the first partial derivatives of the fraction, on the basis of which we construct linear approximations of limit-periodic two-dimensional continued fractions. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Some remarks on a probability limit theorem for continued fractions

It is shown that if a certain condition on the variances of the partial sums is satisfied then a theorem of Philipp and Stout, which implies the asymptotic fluctuation results known for independent random variables, can be applied to some quantities related to continued fractions. Previous results on the behavior of the approximation by the continued fraction convergents to a random real number are improved.

     

Convergence of two-dimensional continued fractions with complex elements

We prove an analog of Vorpits'kii's theorem for two-dimensional continued fractions, applying the formulas for computing their absolute error. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 75–83.     

Optimal continued fractions and the moving average ergodic theorem

We use the moving average ergodic theorem of A. Bellow, R. Jones and J. Rosenblatt to derive various results in metric number theory primarily concerning moving averages of various sequences attached to the optimal continued fraction expansion of a real number.     

On the Van Vleck theorem for limit-periodic continued fractions of general form

J. Koolen posed the problem of studying distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with nonprincipal eigenvalue at most t for a given positive integer t. This problem was solved earlier for t = 3. In the case t = 4, the problem was reduced to studying graphs in which neighborhoods of vertices have parameters (352,26,0,2), (352,36,0,4), (243,22,1,2), (729,112,1,20), (204,28,2,4), (232,33,2,5), (676,108,2,20), (85,14,3,2), or (325,54,3,10). In the present paper, we prove that a distance-regular graph in which neighborhoods of vertices are strongly regular with parameters (85, 14, 3, 2) or (325, 54, 3, 10) has intersection array {85, 70, 1; 1, 14, 85} or {325, 270, 1; 1, 54, 325}. In addition, we find possible automorphisms of a graph with intersection array {85, 70, 1; 1, 14, 85}.     

Problem of interpolation of functions by two-dimensional continued fractions

We investigate the problem of interpolation of functions of two real variables by two-dimensional continued fractions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 842–851, June, 2006.     

Determination of periods of geometric continued fractions for two-dimensional algebraic hyperbolic operators

An explicit construction of a reduced hyperbolic integer operator from the group SL(2, ?) such that one of the periods of the corresponding geometric continued fraction in the sense of Klein coincides with a given sequence of positive integers is presented. An algorithm determining periods for any operator in SL(2, ?) (which is based on Gauss’ reduction theory) is experimentally studied.     

Tasoev's continued fractions and Rogers-Ramanujan continued fractions

We show some new variations on Tasoev's continued fractions , where the periodic parts include the exponentials in k instead of the polynomials in k. We also mention some relations with other kinds of continued fractions, in particular, with Rogers-Ramanujan continued fractions.     

The element sets and value sets of two-dimensional continued fractions

For two-dimensional continued fractions we prove the existence and uniqueness of an optimal sequence of value sets corresponding to an arbitrarily given sequence of element sets. We compute the element set for a given sequence of disk value sets and as a corollary, give the element sets and value sets that are used in convergence criteria for two-dimensional continued fractions. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 39, No. 2, 1996, pp. 55–61.     

A summation formula for divergent continued fractions

Paradoxical formulas which have been proposed by a number of authors for evaluating divergent continued fractions are discussed. The point of the paradoxes is that limit passages (which are to be rigorously defined) result in the convergence real sequences to complex values. These formulas are refined and substantiated on the basis of the theory of uniform distribution supplemented with certain statements of complex analysis.     

A family of continued fractions

We investigate a one-parameter family of infinite generalised continued fractions. The fractions converge to rational values which can be explicitly evaluated. The sequences of numerators and denominators are already to be found in the Online Encyclopedia of Integer Sequences with reference to various topics - but not to continued fractions.     

A generalization of continued fractions

We investigate a generalization of classical continued fractions, where the “numerator” 1 is replaced by an arbitrary positive integer N. We find both similarities to and surprising differences from the classical case.     

On one criterion for the figured convergence of two-dimensional continued fractions with complex elements

We establish a new criterion for the figured convergence of two-dimensional continued fractions with complex elements. This criterion represents a generalization of the theorems of simple and twin convergence sets for continued fractions.