The central limit theorem for Riesz-Raikov sums

Summary Letf be a real-valued function with period 1 satisfying some regularity conditions. It will be proved that, for any ₁,..., _d > 1, the distribution of on the probability space ([0, 1],dt) converges to a normal distribution whose covariance is given by algebraic relations among the _i 's. This generalizes the classical work by M. Kac and refines the characterization off to have a degenerate limit. It also shows that the limit law of is in most cases a Cauchy distribution.     

Continued fractions and the Markoff tree

We give here a full account of Markoff's celebrated result on badly approximable numbers. The proofs rely exclusively on the classical theory of simple continued fractions, together with Harvey Cohn's method using words in the free group with two generators for the determination of the structure of periods of the continued fractions of Markov irrationals. Appendix A gives a short self-contained presentation of the results on continued fractions used here and Appendix B gives short proofs of some results on the still open uniqueness problem for Markoff numbers.     

Maximal continuants and the Fine-Wilf theorem

The following problem was posed by C.A. Nicol: given any finite sequence of positive integers, find the permutation for which the continuant (i.e. the continued fraction denominator) having these entries is maximal, resp. minimal. The extremal arrangements are known for the regular continued fraction expansion. For the singular expansion induced by the backward shift ⌈1/x⌉-1/x the problem is still open in the case of maximal continuants. We present the explicit solutions for sequences with pairwise different entries and for sequences made up of any pair of digits occurring with any given (fixed) multiplicities. Here the arrangements are uniquely described by a certain generalized continued fraction. We derive this from a purely combinatorial result concerning the partial order structure of the set of permutations of a linearly ordered vector. This set has unique extremal elements which provide the desired extremal arrangements. We also prove that the palindromic maximal continuants are in a simple one-to-one correspondence with the Fine and Wilf words with two coprime periods which gives a new analytic and combinatorial characterization of this class of words.     

Shrinking the period length of quasi-periodic continued fractions

Extending the work of Burger et al., here we show that every quasi-periodic simple continued fraction α can be transformed into a quasi-periodic non-simple continued fraction having period length one. Moreover, a certain kind of quasi-periodic non-simple continued fraction is equivalent to a quasi-periodic N-continued fraction. The results of this paper follow from arguments of Burger et al. but we apply our version to offer new continued fractions for certain classes of real numbers.     

A remark on random and equidistributed sequences

The average of the values of a function f on the points of an equidistributed sequence in [0, 1] ^s converges to the integral of f as soon as f is Riemann integrable. Some known low discrepancy sequences perform faster integration than independent random sampling (cf. [1]). We show that a small random absolutely continuous perturbation of an equidistributed sequence allows to integrate bounded Borel functions, and more generally that, if the law of the random perturbation doesn't charge polar sets, such perturbed sequences allow to integrate bounded quasi-continuous functions.     

A recurrence formula for leaping convergents of non-regular continued fractions

Given a continued fraction [a₀;a₁,a₂,…], p_n/q_n=[a₀;a₁,…,a_n] is called the n-th convergent for n=0,1,2,…. Leaping convergents are those of every r-th convergent p_rn+i/q_rn+i (n=0,1,2,…) for fixed integers r and i with r?2 and i=0,1,…,r-1. This leaping step r can be chosen as the length of period in the continued fraction. Elsner studied the leaping convergents p_3n+1/q_3n+1 for the continued fraction of and obtained some arithmetic properties. Komatsu studied those p_3n/q_3n for (s?2). He has also extended such results for some more general continued fractions. Such concepts have been generalized in the case of regular continued fractions. In this paper leaping convergents in the non-regular continued fractions are considered so that a more general three term relation is satisfied. Moreover, the leaping step r need not necessarily to equal the length of period. As one of applications a new recurrence formula for leaping convergents of Apery’s continued fraction of ζ(3) is shown.     

Shrinking the period lengths of continued fractions while still capturing convergents

Here we prove that every real quadratic irrational α can be expressed as a periodic non-simple continued fraction having period length one. Moreover, we show that the sequence of rational numbers generated by successive truncations of this expansion is a sequence of convergents of α. We close with an application relating the structure of a quadratic α to its conjugate.     

Two ergodicity criteria for stochastically recursive sequences

We prove two criteria for so-called coupling- and strong coupling-convergence of stochastically recursive sequences.     

Asymptotic normality of additive functions on polynomial sequences in canonical number systems

The objective of this paper is the study of functions which only act on the digits of an expansion. In particular, we are interested in the asymptotic distribution of the values of these functions. The presented result is an extension and generalization of a result of Bassily and Kátai to number systems defined in a quotient ring of the ring of polynomials over the integers.     

General discrepancy estimates III: The Erdös-Turán-Koksma inequality for the Haar function system

The classical Erdös-Turán-Koksma inequality gives us an upper bound for the discrepancy of a sequence in thes-dimensional unit cube in terms of exponential sums, more precisely, in terms of the trigonometric function system.In this paper, we shall prove the inequality of Erdös-Turán-Koksma for the extreme and the star discrepancy, for generalized Haar function systems. Further, we shall show the existence of the inequality of Erdös-Turán-Koksma for the isotropic discrepancy, for generalized Haar and Walsh function systems.Research supported by the Austrian Science Foundation, project no. P9285/TEC.     

Divisors of Bernoulli Sums

Let where are independent Bernoulli random variables. In relation with the divisor problem, we evaluate the almost sure asymptotic order of the sums , where and is a sequence of positive integers. Received: May 23, 2007. Revised: June 8, 2007.     

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.     

Generalizations of a result of Christensen on renewal sequences and linear recurrences

This note extends some recent work of Christensen on simple closed form solutions for linear recurrences, employing connections with renewal sequences. We obtain results for linear recursive sequences defined via monotone coefficients, and provide a broader result in the third-order case. A related open question is posed.     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.     

A sperner-type theorem

Let P=P ₁×P ₂×...×P _M be the direct product of symmetric chain orders P ₁, P ₂, ..., P _M . Let F be a subset of P containing no l+1 elements which are identical in M–1 components and linearly ordered in the Mth one. Then max |F|cM ^1/2lW(P), where W(P) is the cardinality of the largest level of P, and c is independent of P, M and l. Infinitely many P show that this result is best possible for every M and l apart from the constant factor c.     

A Cassels-Schmidt theorem for non-homogeneous Markov chains

Let r be an integer not less than 2. Suppose that we have a (not necessarily homogeneous) Markov chain with state space {0,1,…,r−1} given by the sequence of r×r transition matrices

     

The numerical analysis on a Volterra equation with asymptotically periodic solution

We consider order one operational quadrature methods on a certain integro-differential equation of Volterra type on (0,∞), with piecewise linear convolution kernels. The forms of discretization solution are patterned after a continuous one of Hannsgen (1979) [2]. An l¹ remainder stability and an error bound are derived.     

On a theorem of Karlin

Starting from an old result of S. Karlin, we demonstrate the usefulness of couplings within the theory of random systems with complete connections. We also give a short exposé of some limit results for the state sequences associated to random systems with complete connections.