在[13]中，K. Roth证明了$N$点van der Corput集的循环移位的$L^2$差异的期望值由$\sqrt{\log N}$的常数倍数限制，从而保证存在具有渐近最小$L^2$差异的移位，[11]。在本文中，我们构建了一个具体的例子，这样的转变。
英文摘要：
In [13], K. Roth showed that the expected value of the $L^2$ discrepancy of the cyclic shifts of the $N$ point van der Corput set is bounded by a constant multiple of $\sqrt{\log N}$, thus guaranteeing the existence of a shift with asymptotically minimal $L^2$ discrepancy, [11]. In the present paper, we construct a specific example of such a shift.
Analytic van der Corput Lemma for p-adic and F_q((t)) oscillatory integrals, singular Fourier transforms, and restriction theorems
本文给出了实vanderCorput引理的p-adic和F_q（（t））模拟，其中相位充分光滑的实条件被相位是收敛的幂级数的条件所取代。这个van der Corput风格的结果，使我们能够在类比的真实情况下，研究奇异的傅里叶变换适当弯曲（解析）流形，并开辟了道路，为进一步的应用。作为这样的进一步应用，我们给出了L^p函数到非阿基米德局部域上适当弯曲的解析流形的傅里叶变换的限制定理，类似于E. Stein和C.弗利曼。
An explicit estimate for the Riemann zeta function on the critical line is derived using the van der Corput method. An explicit van der Corput lemma is presented.
In 1935 J.G. van der Corput introduced a sequence which has excellent uniform distribution properties modulo 1. This sequence is based on a very simple digital construction scheme with respect to the binary digit expansion. Nowadays the van der Corput sequence, as it was named later, is the prototype of many uniformly distributed sequences, also in the multi-dimensional case. Such sequences are required as sample nodes in quasi-Monte Carlo algorithms, which are deterministic variants of Monte Carlo rules for numerical integration. Since its introduction many people have studied the van der Cor…
How to construct generalized van der Corput sequences
作者最近引进的LS-点列是van der Corput序列的推广。它们是通过重新排序划分的相应LS序列的点来构造的。在这里，我们提出了另一种算法，它符合经典的一个van der Corput序列，是简单的计算比原来的建设。该算法基于自然数在基$L+S$中的表示，当$L=b$和S=0时，给出基$b$中的vanderCorput序列.在这种结构中，以及在范德科普特一个，它是必不可少的基础$L+S$中的表示数位的反转：在本文中，我们也给了一个很好的几何解释这个“神奇的”操作。
Let $d_N=ND_N(ω)$ be the discrepancy of the Van der Corput sequence in base $2$. We improve on the known bounds for the number of indices $N$ such that $d_N\leq \log N/100$. Moreover, we show that the summatory function of $d_N$ satisfies an exact formula involving a $1$-periodic, continuous function. Finally, we show that $d_N$ is invariant under digit reversal in base $2$.
在这篇部分说明性的论文中，我们研究了$\Z^d$中的van der Corput集，重点关注与调和分析和测度保持动力系统的递归性质的联系。我们证明了多维版本的一些经典结果$d=1$在\cite{K-MF}和\cite{R}，建立新的特征，介绍和讨论了一些修改的范德科普特集对应于各种概念的复发，提供了大量的例子，并制定一些自然的开放问题。
证明了移位素数集的van der Corput性质的上界为O（（log n）^{-1+o（1）}），从而回答了Ruzsa和Montgomery关于移位素数集p-1的一个问题.本文构造了一个赋范非负值余弦多项式，其谱在p-1，p<=n中，自由系数a_0=O（（logn）^{-1+o（1）}）.这意味着集合p-1的庞加莱性质有相同的界，也意味着与相关集合的均匀分布有关的几个性质有相同的界。
It is well known that the $L_p$-discrepancy for $p \in [1,\infty]$ of the van der Corput sequence is of exact order of magnitude $O((\log N)/N)$. This however is for $p \in (1,\infty)$ not best possible with respect to the lower bounds according to Roth and Proinov. For the case $p=2$ it is well known that the symmetrization trick due to Davenport leads to the optimal $L_2$-discrepancy rate $O(\sqrt{\log N}/N)$ for the symmetrized van der Corput sequence. In this note we show that this result holds for all $p \in (1,\infty)$. The proof is based on an estimate of the Haar coefficients of the co…
We establish Diophantine inequalities for the fractional parts of generalized polynomials $f$, in particular for sequences $ν(n)=\lfloor n^c\rfloor+n^k$ with $c>1$ a non-integral real number and $k\in\mathbb{N}$, as well as for $ν(p)$ where $p$ runs through all prime numbers. This is related to classical work of Heilbronn and to recent results of Bergelson \textit{et al.}
A central limit theorem with explicit error bound, and a large deviation result are proved for a sequence of weakly dependent random variables of a special form. As a corollary, under certain conditions on the function $f: [0,1] \to \mathbb{R}$ a central limit theorem and a large deviation result are obtained for the sum $\sum_{n=0}^{N-1} f(x_n)$, where $x_n$ is the base $b$ van der Corput sequence for an arbitrary integer $b \ge 2$. Similar results are also proved for the $L^p$ discrepancy of the same sequence for $1 \le p < \infty$. The main methods used in the proofs are the Berry-Esseen th…
Variants of the Littlewood conjecture, their connection to uniformly distributed sequences, and the exact order of the discrepancy of van der Corput--Kronecker-type sequences
本文利用有限域上 $X$-进 Littlewood 猜想的反例构造 van der Corput–Kronecker 型序列，并确定其差异度精确阶，从而支持一致分布理论中的最优差异度上界猜想。
Strichartz and dispersive estimates for quantum bouncing ball model: exponential sums and Van der Corput methods in 1d semi-classical Schrödinger equations
本研究针对一维半经典薛定谔方程，通过Van der Corput方法改进Strichartz估计，将损失从1/4降至1/6+ε，并推测高维Friedlander模型亦适用。
This paper establishes a connection between a problem in Potential Theory and Mathematical Physics, arranging points so as to minimize an energy functional, and a problem in Combinatorics and Number Theory, constructing 'well-distributed' sequences of points on $[0,1)$. Let $f:[0,1] \rightarrow \mathbb{R}$ be (i) symmetric $f(x) = f(1-x)$, (ii) twice differentiable on $(0,1)$, and (iii) such that $f''(x)>0$ for all $x \in (0,1)$. We study the greedy dynamical system, where, given an initial set $\{x_0, \ldots, x_{N-1}\} \subset [0,1)$, the point $x_N$ is obtained as $$ x_{N} = \arg\min_{x \in …
The intriguing search for permutations that generate generalised van der Corput sequences with exceptionally small discrepancy forms an important part of the research work of Henri Faure. On the occasion of Henri's 80th birthday we aim to survey (some of) his contributions over the last four decades which considerably improved our understanding of one-dimensional van der Corput sequences and inspired a lot of related work. We recall and compare the different approaches in the search for generalised van der Corput sequences with low discrepancy, i.e., using a single generating permutation versu…
本硕士论文的主要目的是研究由Carbone在\cite{Carbone}中引入的点的$LS$ -序列，并找到它们到单位正方形的两个推广。在这里，我们还提出了一个新的算法提出了同一作者在\cite{Carbone2}，我们实现它，以便有一个图形化的描述这些序列。第1章包括一系列关于均匀分布理论和差异的结果（我们参考\cite{Drmota_Tichy}和\cite{Kuipers_Niederreiter}对这个问题进行全面调查）。在第二章中，我们把我们的注意力集中在$LS$ -序列的划分和点的单位区间，给予特别注意的排序的点“à la van der Corput \rq \rq和找到一种方法来计算它们有关的数字展开的自然数在基$L+S$。在第3章中，我们移动到单位正方形，在那里我们发现了$LS$ -序列的两个推广，这是在多维情况下van der Corput序列的历史发展之后。这就是为什么我们称第一个序列为$LS$ -van der Corput-Hammersley的点序列，第二个序列为$LS$ -Halton的序列。
Van der Corput and Halton sequences are well-known low-discrepancy sequences. Almost twenty years ago Ninomiya defined analogues of van der Corput sequences for $β$-numeration and proved that they also form low-discrepancy sequences if $β$ is a Pisot number. Only very recently Robert Tichy and his co-authors succeeded in proving that $\boldsymbolβ$-adic Halton sequences are equidistributed for certain parameters $\boldsymbolβ=(β_1,\ldots,β_s)$ using methods from ergodic theory. In the present paper we continue this research and give discrepancy estimates for $\boldsymbolβ$-adic Halton sequence…
We use the $q$-analogue of van der Corput's method to estimate short character sums to smooth moduli. If $χ$ is a primitive Dirichlet character modulo a squarefree, $q^δ$-smooth integer $q$ we show that $$L(\frac12,χ)\ll_εq^{\frac{27}{164}+O(δ)+ε}.$$
Let A_N be an N-point distribution in the unit square in the Euclidean plane. We consider the Discrepancy function D_N(x) in two dimensions with respect to rectangles with lower left corner anchored at the origin and upper right corner at the point x. This is the difference between the actual number of points of A_N in such a rectangle and the expected number of points - N x_1x_2 - in the rectangle. We prove sharp estimates for the BMO norm and the exponential squared Orlicz norm of D_N(x). For example we show that necessarily ||D_N||_(expL^2) >c(logN)^(1/2) for some aboslute constant c>0. On …
k Sequences of Generalized Van der Laan and Generalized Perrin Polynomials
本文利用广义Fibonacci多项式和Lucas多项式给出了广义Van der Laan多项式和广义Perrin多项式的k列。我们给出了这些多项式的一些性质。我们还得到了广义k阶Van der Laan数、广义k阶Van der Laan数的k列、广义k阶Perrin数和广义k阶Perrin数的k列。此外，我们还研究了它们之间的关系。
We introduce the van der Waerden complex ${\rm vdW}(n,k)$ defined as the simplicial complex whose facets correspond to arithmetic progressions of length $k$ in the vertex set $\{1, 2, \ldots, n\}$. We show the van der Waerden complex ${\rm vdW}(n,k)$ is homotopy equivalent to a $CW$-complex whose cells asymptotically have dimension at most $\log k / \log \log k$. Furthermore, we give bounds on $n$ and $k$ which imply that the van der Waerden complex is contractible.
