Topic
Van der Corput sequence
About: Van der Corput sequence is a research topic. Over the lifetime, 107 publications have been published within this topic receiving 1961 citations.
Papers published on a yearly basis
Papers
25 Jan 1991
TL;DR: The van der Corput method can be applied to problems such as upper bounds for the Riemann-Zeta function, the Dirichlet divisor problem, the distribution of square free numbers and the Piatetski-Shapiro prime number theorem.
Abstract: This book is a self-contained account of the one- and two-dimensional van der Corput method and its use in estimating exponential sums These arise in many problems in analytic number theory It is the first cohesive account of much of this material and will be welcomed by graduates and professionals in analytic number theory The authors show how the method can be applied to problems such as upper bounds for the Riemann-Zeta function the Dirichlet divisor problem, the distribution of square free numbers, and the Piatetski-Shapiro prime number theorem
606 citations
TL;DR: In this article van der Corput, in connection with his work on distribution functions, was led to the following conjecture which expresses the fact that no sequence can, in a certain sense, be too evenly distributed.
Abstract: In 1935 van der Corput, in connection with his work on distribution functions, was led to the following conjecture which expresses the fact that no sequence can, in a certain sense, be too evenly distributed.
521 citations
TL;DR: It is shown that certain discontinuous functions with infinite variation in the sense of Hardy and Krause can be integrated with a mean squared error of O(n^{-1-1/d}) and certain space-filling curves also attain these rates.
Abstract: Summary
We study the properties of points in generated by applying Hilbert's space filling curve to uniformly distributed points in [0,1]. For deterministic sampling we obtain a discrepancy of for d⩾2. For random stratified sampling, and scrambled van der Corput points, we derive a mean-squared error of for integration of Lipschitz continuous integrands, when d⩾3. These rates are the same as those obtained by sampling on d-dimensional grids and they show a deterioration with increasing d. The rate for Lipschitz functions is, however, the best possible at that level of smoothness and is better than plain independent and identically distributed sampling. Unlike grids, space filling curve sampling provides points at any desired sample size, and the van der Corput version is extensible in n. We also introduce a class of piecewise Lipschitz functions whose discontinuities are in rectifiable sets described via Minkowski content. Although these functions may have infinite variation in the sense of Hardy and Krause, they can be integrated with a mean-squared error of . It was previously known only that the rate was . Other space filling curves, such as those due to Sierpinski and Peano, also attain these rates, whereas upper bounds for the Lebesgue curve are somewhat worse, as if the dimension were times as high.
50 citations
TL;DR: In this paper, the distribution properties of sequences which are a generalization of the well-known van der Corput-Halton sequences on one hand, and digital (T,s)-sequences on the other.
Abstract: We study the distribution properties of sequences which are a generalization of the well-known van der Corput–Halton sequences on one hand, and digital (T,s)-sequences on the other. In this paper, we give precise results concerning the distribution properties of such sequences in the s-dimensional unit cube. Moreover, we consider subsequences of the above-mentioned sequences and study their distribution properties. Additionally, we give discrepancy estimates for some special cases, including subsequences of van der Corput and van der Corput–Halton sequences.
50 citations
TL;DR: In this article, a precise study of the discrepancies between the discrepancies is presented, which is devoted to a precise analysis of the discrepancy between the two sets of data points in the paper.
Abstract: 1. Introduction. This article is devoted to a precise study of the discrepancies D
42 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 2 |
| 2020 | 4 |
| 2019 | 2 |
| 2018 | 5 |
| 2017 | 3 |
| 2016 | 7 |