我们证明了完满平方集合的van der Corput性质的上界是O（（log n）^{-1/3}），从而回答了Ruzsa和Montgomery所考虑的一个问题.我们通过构造非负值的赋范三角多项式来实现，其谱在不超过n的完满平方集合中，并且有一个小的自由系数a_0=O（（log n）^{-1/3}）。
英文摘要：
We prove that the upper bound for the van der Corput property of the set of perfect squares is O((log n)^{-1/3}), giving an answer to a problem considered by Ruzsa and Montgomery. We do it by constructing non-negative valued, normed trigonometric polynomials with spectrum in the set of perfect squares not exceeding n, and a small free coefficient a_0=O((log n)^{-1/3}).
From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules
1935年，J.G. van der Corput引入了一个序列，该序列具有模1的优良均匀分布特性。这个序列是基于一个非常简单的数字构造方案，相对于二进制数字扩展。现在的范德科普特序列，因为它是后来命名的，是许多均匀分布的序列的原型，也在多维的情况下。在拟蒙特卡罗方法中，需要这样的序列作为样本节点，该算法是用于数值积分的蒙特卡罗规则的确定性变体。自从它的介绍，许多人已经研究了范德科普特序列及其推广。这导致了大量的结果。在J.G.的125岁生日之际。vanderCorput我们调查了许多有趣的结果vanderCorput序列及其推广。通过这种方式，我们从van der Corput的思想转移到拟蒙特卡罗方法规则的最现代的序列构造，例如，广义Halton序列或Niederreiter的$(t,s)$ -序列。
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 $2$.
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。
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.}
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。
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。
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。
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…
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。
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…
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 …
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…
As one of the asymptotic formulas of the zeta-function, Hardy and Littlewood gave asymptotic formulas called the approximate functional equation. In this paper, we prove an approximate functional equation of the Barnes double zeta-function $ ζ_2 (s, α; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (α+vm+wn)^{-s} $. Also, applying this approximate functional equation and the van der Corput method, we obtain upper bounds for $ ζ_2(1/2 + it, α; v, w) $ and $ ζ_2(3/2 + it, α; v, w) $ with respect to $ t $ as $ t \rightarrow \infty $.
We give upper bounds for the number of integral solutions of bounded height to a system of equations $f_i(x_1,\ldots,x_n) = 0$, $1 \leq i \leq r$, where the $f_i$ are polynomials with integer coefficients. The estimates are obtained by generalising an approach due to Heath-Brown, using a certain $q$-analogue of van der Corput's method, to the case of systems of polynomials of differing degree. Our results apply for a wider range of $n$, in terms of the degrees of the polynomials $f_i$, than bounds obtained with the circle method.
We study exponential sums of the form $\sum_{n=1}^N e^{2πi a b^n/m}$ for non-zero integers $a,b,m$. Classically, non-trivial bounds were known for $N\ge \sqrt{m}$ by Korobov, and this range has been extended significantly by Bourgain as a result of his and others' work on the sum-product phenomenon. We use a new technique, similar to the Weyl-van der Corput method of differencing, to give more explicit bounds bounds that become non-trivial around the time when $\exp(\log m/\log_2\log m) \le N$. We include applications to the digits of rational numbers and constructions of normal numbers.
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.
