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1.
 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  相似文献   

2.
 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  相似文献   

3.
 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  相似文献   

4.
 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)  相似文献   

5.
For any subset , we introduce the definition of -numbers, which contains the well-known k-free numbers, k-full numbers, k-full and l-free numbers (k<l–1) as special cases. In this paper we study the distribution of -numbers in short intervals. We establish the connection of this problem with the gap problem of k-free numbers and multi-dimensional divisor problems. As applications, we study the short interval distribution of k-full and l-free numbers for k=2, 3, 4, 5, 6, 7.Received June 4, 2002; in revised form January 22, 2003 Published online June 30, 2003  相似文献   

6.
For a positive integer n we let τ(n) denote the number of its positive divisors. In this paper, we obtain lower and upper bounds for the average value of the ratio τ(n + 1)/τ(n) as n ranges through positive integers in the interval [1,x]. We also study the cardinality of the sets {τ(p − 1) : px prime} and {τ(2n − 1) : nx}. Authors’ addresses: Florian Luca, Instituto de Matemáticas, Universidad Nacional Autónoma'de'México, C.P. 58089, Morelia, Michoacán, México; Igor E. Shparlinski, Department of Computing, Macquarie University, Sydney, NSW 2109, Australia  相似文献   

7.
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  相似文献   

8.
 For a real number x let be the fractional part of x and for any set M let c M be the characteristic function of M. For and a positive integer N let
be the discrepancy of the sequence modulo 1. In this paper we prove that
(Received 2 May 2000; in revised form 19 June 2000)  相似文献   

9.
We give conditions sufficient for sequences consisting of real numbers to ensure that their expressible sets have Hausdorff dimension zero.  相似文献   

10.
We prove a general theorem on oscillatory properties of error terms associated with real arithmetical functions whose Mellin transform may have singularities with several summands containing arbitrary complex powers and logarithmic polynomials. The theorem generalizes a theorem of J. Kaczorowski and J. Pintz and is motivated by the applications to algebraic integers with factorizations of distinct lengths. An application of the theorem to the study of the Hilbert semigroup modulo 5 is presented.  相似文献   

11.
In this paper, we consider the mean value of the product of multiplicative arithmetic functions with shifted argument. The investigated functions have to satisfy the following conditions: their moduli do not exceed 1; the values on the set of primes are close to 1 for one of the functions and close to a fixed complex number for the other function. Some consequences for the classical functions are given.  相似文献   

12.
From a finite abelian group G, a quadratic form onG and an element in , we define a topological invariant of a pair(M,L) where is a closed oriented 3-manifold and L an oriented, framedn-component link inM. The main result consists in an explicit formula for this invariant, based on a reciprocity formula for Gauss sums, which features a special linking pairing. This pairing depends on both the quadratic form q and the linking pairing of M. A necessary and sufficient condition for the invariant to vanish is described in terms of a characteristic class for this pairing. We also discuss torsion spin-structures and related structures which appear in this context. Received May 13, 1998 / Accepted November 11, 1999 / Published online February 5, 2001  相似文献   

13.
In this paper, we develop a large sieve type inequality with quadratic amplitude. We use the double large sieve to establish non-trivial bounds.  相似文献   

14.
 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  相似文献   

15.
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  相似文献   

16.
 We define the index of composition λ(n) of an integer n ⩾ 2 as λ(n) = log n/log γ(n), where γ(n) stands for the product of the primes dividing n, and first establish that λ and 1/λ both have asymptotic mean value 1. We then establish that, given any ɛ > 0 and any integer k ⩾ 2, there exist infinitely many positive integers n such that . Considering the distribution function F(z,x) := #{n < x : λ(n) > z}, we prove that, given 1 < z < 2 and ɛ > 0, then, if x is sufficiently large,
this last inequality also holding if z ⩾ 2. We then use these inequalities to obtain probabilistic results and we state a conjecture. Finally, using (*), we show that the probability that the abc conjecture does not hold is 0. Research supported in part by a grant from NSERC. Re?u le 17 décembre 2001; en forme révisée le 23 mars 2002 Publié en ligne le 11 octobre 2002  相似文献   

17.
 We study the asymptotic formula of for some arithmetical functions f and g. This generalizes the case investigated by Balakrishnan and Pétermann. Received 15 January 2001; in revised form 7 July 2001  相似文献   

18.
 We present a method to estimate the L 2-discrepancy of symmetrisized point sets from above and from below with the help of Walsh series analysis. We apply the method to a class of two-dimensional net-type point sets, thereby generalizing results of Halton and Zaremba and of Proinov. (Received 14 September 2000)  相似文献   

19.
 We show that the maximum number of positive integers that appear in k non disjoint paths of the divisorial graph restricted to the numbers ⩽N, is about N log k/log N. We study also some other related questions. Re?u le 20 juin 2000; en forme révisée le 4 Avril 2002  相似文献   

20.
 A link between the theory of error-correcting codes and -nets leads to the efficient construction of two families of very good ternary nets. These have parameters (for all ) and (for all odd ). The underlying codes are cyclic codes. (Received 26 June 2000; in revised form 2 October 2000)  相似文献   

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