On the discrepancy of (0,1)-sequences

We give bounds for the L^p-discrepancy, , of the van der Corput sequence in base 2. Further, we give a best possible upper bound for the star discrepancy of (0,1)-sequences and show that this bound is attained for the van der Corput sequence. Finally, we give a (0,1)-sequence with essentially smaller star discrepancy than for the van der Corput sequence.     

Average growth-behavior and distribution properties of generalized weighted digit-block-counting functions

We introduce a generalized weighted digit-block-counting function on the nonnegative integers, which is a generalization of many digit-depending functions as, for example, the well known sum-of-digits function. A formula for the first moment of the sum-of-digits function has been given by Delange in 1972. In the first part of this paper we provide a compact formula for the first moment of the generalized weighted digit-block-counting function and show that a (weak) Delange type formula holds if the sequence of weights converges. The question, whether the converse is true as well, can only be answered partially at the moment. In the second part of this paper we study distribution properties of generalized weighted digit-block-counting sequences and their d-dimensional analogues. We give an if and only if condition under which such sequences are uniformly distributed modulo one. Roswitha Hofer, Recipient of a DOC-FFORTE-fellowship of the Austrian Academy of Sciences at the Institute of Financial Mathematics at the University of Linz (Austria). Friedrich Pillichshammer, Supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”. Dedicated to Prof. Robert F. Tichy on the occasion of his 50th birthday Authors’ address: Roswitha Hofer, Gerhard Larcher and Friedrich Pillichshammer, Institut für Finanzmathematik, Universit?t Linz, Altenbergerstra?e 69, A-4040 Linz, Austria     

Arithmetical properties of linear recurrent sequences

Let F(z)∈R[z] be a polynomial with positive leading coefficient, and let α>1 be an algebraic number. For r=degF>0, assuming that at least one coefficient of F lies outside the field Q(α) if α is a Pisot number, we prove that the difference between the largest and the smallest limit points of the sequence of fractional parts {F(n)αn}_n=1,2,3,… is at least 1/?(Pr+1), where ? stands for the so-called reduced length of a polynomial.     

On the -Discrepancy of the Hammersley Point Set

 We give a formula for the -discrepancy of the 2-dimensional Hammersley point set in base 2 for all integers p, . Received 18 May 2001; in revised form 18 December 2001     

Uniformly maldistributed sequences in a strict sense

It is shown that the following three limits

Mahler measures in a field are dense modulo 1

Let K be a number field. We prove that the set of Mahler measures M(α), where α runs over every element of K, modulo 1 is everywhere dense in [0, 1], except when or , where D is a positive integer. In the proof, we use a certain sequence of shifted Pisot numbers (or complex Pisot numbers) in K and show that the corresponding sequence of their Mahler measures modulo 1 is uniformly distributed in [0, 1]. Received: 24 March 2006     

The great theorem of A.A. Markoff and Jean Bernoulli sequences

A proof of Markoff’s Great Theorem on the Lagrange spectrum using continued fractions is sketched. Markoff’s periods and Jean Bernoulli sequence ¹ are used to obtain a simple algorithm for the computation of the Lagrange spectrum below 3.     

The distribution of patterns of inverses modulo a prime

We investigate the distribution of the numbers x∈[1,p] for which all lie in a subset of the set of multiplicative inverses modulo a prime p. Here the a_i are integers coprime to p and the numbers are distinct .     

On the number of lattice points between two enlarged and randomly shifted,copies of an oval

Summary Let A be an oval with a nice boundary in ²,R a large positive number,c>0 some fixed number and a uniformly distributed random vector in the unit square [0,1]². We are interested in the number of lattice points in the shifted annular region consisting of the difference of the sets {(R+c/R)A–} and {(R–c/R)A–}. We prove that whenR tends to infinity, the expectation and the variance of this random variable tend to 4c times the area of the set A, i.e. to the area of the domain where we are counting the number of lattice points. This is consistent with computer studies in the case of a circle or an ellipse which indicate that the distribution of this random variable tends to the Poisson law. We also make some comments about possible generalizations.     

On Strong Uniform Distribution II

 Let ? be a class of real valued integrable functions on [0,1). We will call a strictly increasing sequence of natural numbers an sequence if for every f in ? we have

Generalized radix representations and dynamical systems. IV

For r = (r₁,…, r_d) ∈ ?^d the mapping τ_r:?^d →?^d given byτ_r(a₁,…,a_d) = (a₂, …, a_d,−⌊r₁a₁+…+ r_da_d⌋)where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ?^d there exists an integer k > 0 with τ_r^k(a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).     

On the proof of humbert’s volume formula

We close a gap in Humbert’s classical calculation of the volume of the quotient of three-dimensional hyperbolic space by SL₂ over the ring of integers of an imaginary quadratic number field.     

The tent transformation can improve the convergence rate of quasi-Monte Carlo algorithms using digital nets

In this paper we investigate multivariate integration in reproducing kernel Sobolev spaces for which the second partial derivatives are square integrable. As quadrature points for our quasi-Monte Carlo algorithm we use digital (t,m,s)-nets over which are randomly digitally shifted and then folded using the tent transformation. For this QMC algorithm we show that the root mean square worst-case error converges with order for any ɛ > 0, where 2 ^m is the number of points. A similar result for lattice rules has previously been shown by Hickernell. Ligia L. Cristea is supported by the Austrian Research Fund (FWF), Project P 17022-N 12 and Project S 9609. Josef Dick is supported by the Australian Research Council under its Center of Excellence Program. Gunther Leobacher is supported by the Austrian Research Fund (FWF), Project S 8305. Friedrich Pillichshammer is supported by the Austrian Research Fund (FWF), Project P 17022-N 12, Project S 8305 and Project S 9609.     

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.     

Unimodular eigenvalues, uniformly distributed sequences and linear dynamics

We study increasing sequences of positive integers (nk)_k?1 with the following property: every bounded linear operator T acting on a separable Banach (or Hilbert) space with supk?1‖Tnk‖<∞ has a countable set of unimodular eigenvalues. Whether this property holds or not depends on the distribution (modulo one) of sequences (nkα)_k?1, α∈R, or on the growth of nk+1/nk. Counterexamples to some conjectures in linear dynamics are given. For instance, a Hilbert space operator which is frequently hypercyclic, chaotic, but not topologically mixing is constructed. The situation of C₀-semigroups is also discussed.     

Some remarks on countably determined measures and uniform distribution of sequences

Two topics are investigated: countably determined (regular Borel probability) measures on compact Hausdorff spaces, and uniform distribution of sequences regarding mainly this kind of measures. We prove several characterizations of countably determined measures, and apply the results in order to show the existence of a well distributed sequence in the support of a countably determined measure. We also generalize a result of Losert on the existence of uniformly distributed sequences in compact dyadic spaces.     

The joint value distribution of the Riemann zeta function and Hurwitz zeta functions II

In the previous paper [9] the author proved the joint limit theorem for the Riemann zeta function and the Hurwitz zeta function attached with a transcendental real number. As a corollary, the author obtained the joint functional independence for these two zeta functions. In this paper, we study the joint value distribution for the Riemann zeta function and the Hurwitz zeta function attached with an algebraic irrational number. Especially we establish the weak joint functional independence for these two zeta functions. Received: 17 Apri1 2007     

Construction of large families of pseudorandom binary sequences

In a series of papers Mauduit and Sárközy (partly with coauthors) studied finite pseudorandom binary sequences. They showed that the Legendre symbol forms a “good” pseudorandom sequence, and they also tested other sequences for pseudorandomness, however, no large family of “good” pseudorandom sequences has been found yet.In this paper, a large family of this type is constructed by extending the earlier Legendre symbol construction.     

On the error term in the mean square formula for the Riemann zeta-function in the critical strip

For 1/2<<1 fixed, letE (T) denote the error term in the asymptotic formula for . We obtain some new bounds forE (T), and an _-result which is the analogue of the strongest _-result in the classical Dirichlet divisor problem.     

On the best bound of the minimal twisted height of linear subspaces

Let V be a vector space over a global field k, g an element of the adele group and H_g the twisted height defined on the k-subspaces of V . We show that the square root of the generalized Hermite-Rankin constant for k gives the best upper bound of the function , where runs over all m-dimensional k-subspaces of V and runs over all n-dimensional k-subspaces of . Received: 17 June 2005