On Strong Uniform Distribution,III

We construct infinite dimensional chains that are ¹ good for almost sure convergence, which settles a question raised in this journal [7] and earlier in [6] by R. Nair. In [7] it was stated that the construction proposed in [4] was invalid. We complete the construction proposed in [4], where it is true that a piece of proof was forgotten. The technic remains the same and the completion of the proof rather natural.     

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

On Convergence Rate in the Gauss–Kuzmin Problem for Grotesque Continued Fractions

 We give an infinite-order-chain representation of the sequence of the incomplete quotients of the grotesque continued fraction expansion. Together with the ergodic behaviour of a certain homogeneous random system with complete connections, this allows us to prove a Gauss–Kuzmin-type theorem for this expansion. Finally, we derive a two-dimensional Gauss–Kuzmin theorem and also obtain an estimate of the convergence rate. (Received 9 October 2000; in revised form 27 March 2001)     

An iterated logarithm law related to decimal and continued fraction expansions

For an irrational number x and n ≥ 1, we denote by k _n (x) the exact number of partial quotients in the continued fraction expansion of x given by the first n decimals of x. G. Lochs proved that for almost all x, with respect to the Lebesgue measure In this paper, we prove that an iterated logarithm law for {k _n (x): n ≥ 1}, more precisely, for almost all x, for some constant σ > 0. Author’s address: Department of Mathematics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R. China     

Lattice Points in Planar Convex Domains

We investigate the number of lattice points in planar convex domains. We give estimates of the remainder in the asymptotic representation with numerical constants, which are astonishingly small. We consider convex planar domains whose boundary has nonvanishing curvature throughout. Here the curvature of the curve of boundary plays an important role. Further, we consider the number of lattice points in domains which are bounded by superellipses. These curves have isolated points with curvature zero.     

On Some Exponential Diophantine Equations

 We apply a new, deep result of Bilu, Hanrot and Voutier to solve completely some exponential Diophantine equations of the type , where are given coprime positive integers, , and are unknown. (Received 29 May 2000; in revised form 21 September 2000)     

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     

On Powerful Normal Subgroups of a p-Group

 In this paper we investigate the action of a p-group G on its powerful normal subgroup N. We shall mainly be concerned to find conditions which guarantee that G acts uniserially on N. We shall also study what consequences uniserial action on N has on the structure of N. (Received 21 August 1998; in revised form 28 June 1999)     

The Existence of Good Extensible Polynomial Lattice Rules

Extensible (polynomial) lattice rules have been introduced recently and they are convenient tools for quasi-Monte Carlo integration. It is shown in this paper that for suitable infinite families of polynomial moduli there exist generating parameters for extensible rank-1 polynomial lattice rules such that for all these infinitely many moduli and all dimensions s the quantity R ^(s) and the star discrepancy are small. The case of Korobov-type polynomial lattice rules is also considered.Received April 30, 2002; in revised form August 21, 2002 Published online April 4, 2003     

On the Critical Dimension and AC Entropy for Markov Odometers

Markov odometers are natural models for non-homogeneous Markov chains, and are natural generalisations of infinite product measures. We show how to calculate the critical dimension of these measures: this is an invariant which describes the asymptotic growth rate of sums of Radon-Nikodym derivatives. This interesting invariant appears to give a kind of entropy for non-singular odometer actions. The techniques require a law of large numbers for inhomogeneous Markov chains.     

Lattice Points in Convex Bodies with Planar Points on the Boundary

 An asymptotic formula is proved for the number of lattice points in large threedimensional convex bodies. In contrast to the usual assumption the Gaussian curvature of the boundary may vanish at non-isolated points. It is only assumed that the second fundamental form vanishes at isolated points where the tangent plane is rational and some ellipticity condition holds. Received 25 April 2001     

Partial Densities on Locally Compact Abelian Groups and Uniformly Distributed Sequences

 We investigate conditions under which a partial density on a locally compact abelian group can be extended to a density. The results allow applications to the theory of uniform distribution of sequences in locally compact abelian groups. Received August 27, 2001 Published online July 12, 2002     

Lattice Points in Three-Dimensional Convex Bodies with Points of Gaussian Curvature Zero at the Boundary

 In this article we investigate the number of lattice points in a three-dimensional convex body which contains non-isolated points with Gaussian curvature zero but a finite number of flat points at the boundary. Especially, in case of rational tangential planes in these points we investigate not only the influence of the flat points but also of the other points with Gaussian curvature zero on the estimation of the lattice rest. Received 19 June 2001; in revised form 17 January 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Cubic Fields with Index 2

 Let d be a squarefree integer with allowed. If (mod 8) it is shown that there do not exist any cubic fields with index 2 whose splitting field contains . If (mod 8) it is shown that there exist infinitely many cubic fields with index 2 and minimal index 2 whose splitting field contains . (Received 23 May 2001)     

On Le’s and Bugeaud’s Papers about the Equation ax 2?+ b 2m?1 ?=?4 c p

 We correct an inaccuracy in the papers of Le, Mignotte, and Bugeaud on the generalized Ramanujan-Nagell equation. Received June 23, 2001     

Metric Theorems for Distribution Measures of Pseudorandom Sequences

 We improve a recent result of Mauduit and Sárk?zy (2000) on the well-distribution measure of pseudorandom sequences and complement another of their results on the correlation measure. Received 12 September 2000; in revised form 1 March 2001     

Expressible sets of sequences with Hausdorff dimension zero

We give conditions sufficient for sequences consisting of real numbers to ensure that their expressible sets have Hausdorff dimension zero.     

Construction of pseudorandom binary lattices by using the multiplicative inverse

In an earlier work Hubert and the authors of this paper introduced and studied the notion of pseudorandomness of binary lattices. Later in another paper the authors gave a construction for a large family of “good” binary lattices by using the quadratic characters of finite fields. Here, a further large family of “good” binary lattices is constructed by using finite fields and the notion of multiplicative inverse. Authors’ addresses: Christian Mauduit, Institut de Mathématiques de Luminy, CNRS, UMR 6206, 163 avenue de Luminy, Case 907, F-13288 Marseille Cedex 9, France; András Sárk?zy, Department of Algebra and Number Theory, E?tv?s Loránd University, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary     

Entropy conditions for subsequences of random variables with applications to empirical processes

We introduce new entropy concepts measuring the size of a given class of increasing sequences of positive integers. Under the assumption that the entropy function of is not too large, many strong limit theorems will continue to hold uniformly over all sequences in . We demonstrate this fact by extending the Chung-Smirnov law of the iterated logarithm on empirical distribution functions for independent identically distributed random variables as well as for stationary strongly mixing sequences to hold uniformly over all sequences in . We prove a similar result for sequences (n _k ω) mod 1 where the sequence (n _k ) of real numbers satisfies a Hadamard gap condition. Authors’ addresses: István Berkes, Department of Statistics, Technical University Graz, Steyrergasse 17/IV, A-8010 Graz, Austria; Walter Philipp, Department of Statistics, University of Illinois, 725 S. Wright Street, Champaign, IL 61820, USA; Robert F. Tichy, Department of Analysis and Computational Number Theory, Technical University Graz, Steyrergasse 30, A-8010 Graz, Austria     

On the Metrical Theory of Continued Fraction Mixing Fibred Systems and Its Application to Jacobi-Perron Algorithm

 We study the metrical theory of fibred systems, in particular, in the case of continued fraction mixing systems. We get the limit distribution of the largest value of a continued fraction mixing stationary stochastic process with infinite expectation and some related results. These are analogous to J. Galambos, W. Philipp, and H. G. Diamond–J. D. Vaaler theorems for the regular continued fractions. As an application, we see that these theorems hold for Jacobi-Perron algorithm. Received September 30, 2001; in revised form January 8, 2002