On the Discrepancy for Boxes and Polytopes

 We consider so-called Tusnády’s problem in dimension d: Given an n-point set P in R ^d , color the points of P red or blue in such a way that for any d-dimensional interval B, the number of red points in differs from the number of blue points in by at most Δ, where should be as small as possible. We slightly improve previous results of Beck, Bohus, and Srinivasan by showing that , with a simple proof. The same asymptotic bound is shown for an analogous problem where B is allowed to be any translated and scaled copy of a fixed convex polytope A in R ^d . Here the constant of proportionality depends on A and we give an explicit estimate. The same asymptotic bounds also follow for the Lebesgue-measure discrepancy, which improves and simplifies results of Beck and of Károlyi. Received 17 November 1997; in revised form 30 July 1998     

On the Average Discrepancy of Successive Tuples of Pseudo-Random Numbers over Parts of the Period

 In the present paper we give an upper and a lower bound for the average value of the discrepancy of non-overlapping s-tuples of successive elements of a first order congruential pseudo-random-number generator (with prime modulus and maximal period). The estimates are – up to logarithmic factors – sharp also for short parts of the period. Received 30 January 1997; in revised form 2 May 1997     

Discrepancy Estimates on the Sphere

 For measures on the unit sphere in ℝ ^d , d≥3, we derive discrepancy estimates in terms of the quality of corresponding quadrature formulas and in terms of bounds for potential differences. (Received 1 August 1998; in revised form 30 December 1998)     

On the Discrepancy of Point Distributions on Spheres and Hyperbolic Spaces

 Optimal lower bounds are given for the discrepancy of point distributions w.r.t. geodesic balls on spheres and hyperbolic spaces. The mean discrepancy is estimated below by using a non-commutative version of the Fourier transform method developed by Beck for Euclidean spaces. Received March 2, 2000; in revised form February 28, 2002 Published online August 19, 2002     

On Finite Pseudorandom Binary Sequences, VI,(On Sequences)

 Let K be a positive integer and α be a real number, and for let if the fractional part of is , and if it is . The pseudorandom properties of the sequence are studied. As measures of pseudorandomness, the regularity of the distribution relative to arithmetic progressions and the correlation are used. In a previous paper the authors studied the special cases and , while here the case is considered. (Received 5 November 1997; in revised form 7 March 2000)     

Involutorische Automorphismen ebener konvexer Mengen

 A bijection of the compact convex set with is called a reflection of K, if σ maps convex subsets of K into convex subsets. Conditions are stated unter which the existence of few reflections imply that is an ellipse. Received 4 March, 1998     

Approximation of Convex Bodies and a Momentum Lemma for Power Diagrams

 The volume of the symmetric difference of a smooth convex body in and its best approximating polytope with n vertices is asymptotically a constant multiple of . We determine this constant and the similarly defined constant for approximation with a given number of facets by solving two isoperimetric problems for planar tilings. Received 15 May 1997; in revised form 14 August 1997     

Some Congruences Modulo Primes

 We shall prove a sequence of congruences modulo odd primes p which can be viewed as generalizations of a congruence first proved by Zhi-Wei Sun in 1995. Received 2 December 1997     

On the Density of 2-Saturated Lattice Packings of Discs

 In a recent paper of G. Fejes Tóth, G. Kuperberg and W. Kuperberg [1] a conjecture has been published concerning the greatest lower bound of the density of a 2-saturated packing of unit discs in the plane. (A packing of unit discs is said to be 2-saturated if none of the discs could be replaced by two other ones of the same size to generate a new packing. A packing of the unit disc is a lattice packing if the centers form a point lattice.) In the present note we study this problem for lattice packings, however, in a more general form in which the removed unit disc is replaced by two discs of radius r. A corollary of our results supports the above conjecture proving that a lattice packing cannot be 2-saturated except if its density is larger than the conjectured bound. (Received 6 December 2000; in revised form March 29, 2001)     

Invertibility Conditions and Bounds for Spectra of Matrix Integral Operators

 Matrix integral operators are considered. Bounds for the spectrum are established. In particular, they give the invertibility conditions and estimates for the spectral radius. (Received 16 December 1998; in revised form 30 May 1999)     

Discrepancy of Fractions with Divisibility Constraints

We provide bounds for the absolute discrepancy of sequences of fractions with denominators streaming in a given arithmetic progression and satisfying divisibility constraints. Supported by the CERES Program 4-147/2004 of the Romanian Ministry of Education and Research.     

Inequalities for Convex Hulls of Random Points

 We prove an estimate for the probability that the convex hull of j independent random points is disjoint from the convex hull of k further independent random points chosen in a plane convex body. (Received 25 January 2000)     

On Hardy-Type Inequalities and Hankel Transforms

 For the real Hardy spaces , we shall show the Hardy type integral inequalities, and applying the inequalities we shall establish the Hardy’s inequalities with respect to Hankel transforms. Received 21 October 1997; in revised form 19 October 1998     

The View-Obstruction Problem for 5-Dimensional Cubes

 The view-obstruction problem for the n-dimensional cube is equivalent to the conjecture that for any n positive integers there is a real number x such that each (here denotes the distance from y to the nearest integer). This conjecture has been previously solved for . In this paper we prove that when we can find x which gives each ; this is the first improvement over the easy result . Received 5 August 1997; in revised form 19 January 1998     

Universal Köthe Sequence Spaces

 A characterization is given for the K?the matrices B such that the K?the sequence space , with , contains all K?the sequence spaces of order p as subspaces. It follows that the class of K?the sequence spaces of order p has a universal element which is quasinormable. In particular, there is a quasinormable space (respectively, which contains every nuclear Fréchet space with basis (respectively, every countably normed Fréchet Schwartz space). Only Fréchet spaces with continuous norm are considered in this note. Received 15 January 1997; in final form 9 June 1997     

Sur l’ordre de grandeur des quotients partiels du développement en fractions continues régulières

 For any irrational , let denote the regular continued fraction expansion of x and define f, for all z > 0 by and by J. GALAMBOS proved that (μ the Gauss measure)

On Finite Pseudorandom Binary Sequences, V.On (nα) and (n 2α) Sequences

 Let k be a positive integer and α be a real number, and for if the fractional part of is <1/2 and e _n =−1 if it is ≥1/2. The pseudorandom properties of the sequence are studied. As measures of pseudorandomness, the regularity of the distribution relative to arithmetic progressions and the correlation are used. Here the special cases k=1 and k=2 are studied (while the case k>2 will be studied in the sequel). (Received 23 April 1999)     

On the Multidimensional Distribution of Inversive Congruential Pseudorandom Numbers in Parts of the Period

 The inversive congruential method is an attractive alternative to the classical linear congruential method for pseudorandom number generation. In this paper we present the first nontrivial bounds on the multidimensional discrepancy of individual sequences of inversive congruential pseudorandom numbers in parts of the period. The proof is based on a new bound for certain incomplete exponential sums. (Received 3 December 1998)     

Discrepancy and the Error in Integration

 This paper is devoted to an estimation of the error of integration with respect to arbitrary unit measures μ and ν on only in terms of continuity or smoothness properties of the function f and the discrepancy . Here, stands for certain classes of (Borel-) test sets. The proofs are in part based on a continuous wavelet analysis of the integrated function by means of Haar-type wavelets. Received 26 January 2001; in revised form 23 September 2001