A new infinite product representation for real numbers

We introduce a new algorithm that leads to a representation for any real number greater than one as an infinite product of rational numbers. Just as we can regard the Cantor product as being a product analogue of the series of Sylvester, this new product is analogous to the classical Engel representation for real numbers. The growth conditions satisfied by the digits in the product are likewise shown to correspond to those required for the Engel series. The representation for certain types of rational numbers via this algorithm is also considered.     

Frequency of digits in the Lüroth expansion

In this note we consider the Lüroth expansion of a real number, and we study the Hausdorff dimension of a class of sets defined in terms of the frequencies of digits in the expansion. We also study the speed at which the approximants obtained from the Lüroth expansion converge. In addition, we describe the multifractal properties of the level sets of the Lyapunov exponent, which measures the exponential speed of approximation obtained from the approximants. Finally, we describe the relation of the Lüroth expansion with the continued fraction expansion and the β-expansion. We remark that our work is still another application of the theory of dynamical systems to number theory.     

On the error-sum function of Lüroth series

We introduce the error-sum function of Lüroth series. Some elementary properties of this function are studied. We also determine the Hausdorff dimension of the graph of this function.     

On the error-sum function of tent map base series

We first introduce tent map base series. The tent map base series is special case of generalized Lüroth series which has the tent map as a base map. Then we study some elementary properties of its error-sum function, and show that the function is continuous.     

q-ENGEL SERIES EXPANSIONS AND SLATER'S IDENTITIES

Abstract

Dedicated to the memory of John Knopfmacher (1937–1999)

We describe the q-Engel series expansion for Laurent series discovered by John Knopfmacher and use this algorithm to shed new light on partition identities related to two entries from Slater's list. In our study Al-Salam/Ismail and Santos polynomials play a crucial r?ole.     

On new fractal phenomena connected with infinite linear IFS

We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. We show that the systems of cylinders of generalized Lüroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' approach, we prove sufficient conditions for the non‐faithfulness of such families of cylinders. On the other hand, rather general sufficient conditions for the faithfulness of such covering systems are also found. As a corollary, we obtain the non‐faithfullness of the family of cylinders generated by the classical Lüroth expansion. We also develop new approach to the study of subsets of ‐essentially non‐normal numbers and prove that this set has full Hausdorff dimension. This result answers the open problem mentioned in 2 and completes the metric, dimensional and topological classification of real numbers via the asymptotic behaviour of frequencies their digits in the generalized Lüroth expansion.     

On the Lebesgue measure of sum-level sets for Lüroth expansion

In this paper the sum-level sets for Lüroth expansion are introduced. We prove that the Lebesgue measure of these sum-level sets decays to zero as the level tends to infinity.     

Gedämpfte zahlentheoretische Transformationen

Generalizing earlier work ofJ. Galambos and the present author a model of a class of transformations mapping the unit interval into itself is proposed. This class contains the classical series expansions of Sylvester and Engel, the algorithm of Pierce and a lot of their genralizations. The main result is an ergodic theorem: the space mean of a certain sequence equals the time mean almost every-where.     

On the group extension of the transformation associated to non-archimedean continued fractions

Summary Let F_q be a finite field with q elements. We consider formal Laurent series of F_q -coefficients with their continued fraction expansions by F_q -polynomials. We prove some arithmetic properties for almost every formal Laurent series with respect to the Haar measure. We construct a group extension of the non-archimedean continued fraction transformation and show its ergodicity. Then we get some results as an application of the individual ergodic theorem. We also discuss the convergence rate for limit behaviors.     

A remark on a theorem of Bavula on Lüroth extensions

Using a theorem of Roquette-Ohm [P. Roquette, Isomorphisms of generic splitting fields of simple algebras, J. Reine Angew. Math. 214-215 (1964) 207-226 and J. Ohm, On subfields of rational function fields, Arch. Math. 42 (1984) 136-138], we indicate a short proof that a rational extension of a Lüroth extension of a field k is a Lüroth extension of k. This assertion was proved by Bavula [V. Bavula, Lüroth field extensions, J. Pure Appl. Algebra 199 (2005) 1-10] with the added hypothesis that .     

KNOPFMACHER EXPANSIONS IN NUMBER THEORY

Abstract

Dedicated to the memory of John Knopfmacher (1937–1999)

We survey in this paper J. Knopfmacher results on number theoretical expansions as they arised from our common collaboration.     

ON THE ERROR-SUM FUNCTION OF ALTERNATING LǖIROTH SERIES

The error-sum function of alternating Lǖroth series is introduced, which, to some extent, discerns the superior or not of an expansion comparing to other expansions. Some elementary properties of this function are studied. Also, the Hausdorff dimension of graph of such function is determined.     

An analogue of a theorem of Szüsz for formal Laurent series over finite fields

About 40 years ago, Szüsz proved an extension of the well-known Gauss-Kuzmin theorem. This result played a crucial role in several subsequent papers (for instance, papers due to Szüsz, Philipp, and the author). In this note, we provide an analogue in the field of formal Laurent series and outline applications to the metric theory of continued fractions and to the metric theory of diophantine approximation.     

Metric properties and exceptional sets of β-expansions over formal Laurent series

This paper is concerned with the metric properties of β-expansions over the field of formal Laurent series. We will see that there are essential differences between β-expansions of the formal Laurent series case and the classical real case. Also the Hausdorff dimensions of some exceptional sets, with respect to the Haar measure, are determined.     

Unexpected distribution phenomenon resulting from Cantor series expansions

We explore in depth the number theoretic and statistical properties of certain sets of numbers arising from their Cantor series expansions. As a direct consequence of our main theorem we deduce numerous new results as well as strengthen the known ones.     

Diagonals of the laurent series of rational functions

We consider the problem of the algebraicity of diagonal series for the Laurent expansions of rational functions, geometrically identifiable using the amoeba of the denominator or an integer point in its Newton polyhedron. We give sufficient conditions for the algebraicity of diagonals basing on the theory of multidimensional residues and topological properties of the complements to collections of complex hypersurfaces in complex analytic varieties.     

Expansion of Weighted Pseudoinverse Matrices in Matrix Power Products

On the basis of the Euler identity, we obtain expansions for weighted pseudoinverse matrices with positive-definite weights in infinite matrix power products of two types: with positive and negative exponents. We obtain estimates for the closeness of weighted pseudoinverse matrices and matrices obtained on the basis of a fixed number of factors of matrix power products and terms of matrix power series. We compare the rates of convergence of expansions of weighted pseudoinverse matrices in matrix power series and matrix power products to weighted pseudoinverse matrices. We consider problems of construction and comparison of iterative processes of computation of weighted pseudoinverse matrices on the basis of the obtained expansions of these matrices.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1539–1556, November, 2004.     

Free continuous actions on zero-dimensional spaces

We show that every countably infinite group admits a free, continuous action on the Cantor set having an invariant probability measure. We also show that every countably infinite group admits a free, continuous action on a non-homogeneous compact metric space and the action is minimal (that is to say, every orbit is dense). In answer to a question posed by Giordano, Putnam and Skau, we establish that there is a continuous, minimal action of a countably infinite group on the Cantor set such that no free continuous action of any group gives rise to the same equivalence relation.     

Expansions of one density via polynomials orthogonal with respect to the other

We expand the Chebyshev polynomials and some of its linear combination in linear combinations of the q-Hermite, the Rogers (q-utraspherical) and the Al-Salam-Chihara polynomials and vice versa. We use these expansions to obtain expansions of some densities, including q-Normal and some related to it, in infinite series constructed of the products of the other density times polynomials orthogonal to it, allowing deeper analysis and discovering new properties. On the way we find an easy proof of expansion of the Poisson-Mehler kernel as well as its reciprocal. We also formulate simple rule relating one set of orthogonal polynomials to the other given the properties of the ratio of the respective densities of measures orthogonalizing these polynomials sets.     

ON THE ERROR-SUM FUNCTION OF ALTERNATING L(U)ROTH SERIES

The error-sum function of alternating Lüroth series is introduced, which, to some extent, discerns the superior or not of an expansion comparing to other expansions. Some elementary properties of this function are studied. Also, the Hausdorff dimension of graph of such function is determined.