The LS-sequences of points recently introduced by the author are a generalization of van der Corput sequences. They were constructed by reordering the points of the corresponding LS-sequences of partitions. Here we present another algorithm which coincides with the classical one for van der Corput sequences and is simpler to compute than the original construction. This algorithm is based on the representation of natural numbers in base $L+S$ and gives the van der Corput sequence in base $b$ if $L=b$ and S=0. In this construction, as well as in the van der Corput one, it is essential the inversion of digits of the representation in base $L+S$: in this paper we also give a nice geometrical explanation of this "magical" operation.
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…
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}).
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 $2$.
We prove that the upper bound for the van der Corput property of the set of shifted primes is O((log n)^{-1+o(1)}), giving an answer to a problem considered by Ruzsa and Montgomery for the set of shifted primes p-1. We construct normed non-negative valued cosine polynomials with the spectrum in the set p-1, p<=n, and a small free coefficient a_0=O((log n)^{-1+o(1)}). This implies the same bound for the Poincaré property of the set p-1, and also bounds for several properties related to uniform distribution of related sets.
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 improve the error term in the van der Corput transform for exponential sums \sum_{a \le n \le b} g(n) exp(2πi f(n)). For many functions g and f, we can extract the next term in the asymptotic, showing that previous results, such as those of Karatsuba and Korolev, are sharp. Of particular note, the methods of this paper avoid the use of the truncated Poisson formula, and thus can be applied to much longer intervals [a,b] with far better results. We provide a detailed analysis of the error term in the case g(x)=1 and f(x)=(x/3)^{3/2}.
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.}
Similarly to $β$-adic van der Corput sequences, abstract van der Corput sequences can be defined for abstract numeration systems. Under some assumptions, these sequences are low discrepancy sequences. The discrepancy function is computed explicitely, and a characterization of bounded remainder sets of the form $[0,y)$ is provided.
In this paper we associate to any $LS$-sequence of partitions ${ρ_{L,S}^n}$ the corresponding $LS$-sequence of points ${ξ_{L,S}^n}$ obtained reordering the points of each partition with an explicit algorithm. The procedure begins with the representation in base $L+S$ of natural numbers, $[n]_{L+S}$, and ends with the $LS$-radical inverse function $ϕ_{L,S}$, introduced ad hoc, evaluated at an appropriate subsequence of natural numbers depending on $L$ and $S$. This construction is deeply related to the geometric representation of the points of ${ξ_{L,S}^n}$ by suitable affine functions and remi…
In this partly expository paper we study van der Corput sets in $\Z^d$, with a focus on connections with harmonic analysis and recurrence properties of measure preserving dynamical systems. We prove multidimensional versions of some classical results obtained for $d=1$ in \cite{K-MF} and \cite{R}, establish new characterizations, introduce and discuss some modifications of van der Corput sets which correspond to various notions of recurrence, provide numerous examples and formulate some natural open questions.
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.
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。
We give the p-adic and F_q((t)) analogue of the real van der Corput Lemma, where the real condition of sufficient smoothness for the phase is replaced by the condition that the phase is a convergent power series. This van der Corput style result allows us, in analogy to the real situation, to study singular Fourier transforms on suitably curved (analytic) manifolds and opens the way for further applications. As one such further application we give the restriction theorem for Fourier transforms of L^p functions to suitably curved analytic manifolds over non-archimedean local fields, similar to …
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…
Let $K\subset\mathbb N$ and $\mathbf T(K)$ is a set of trigonometric polynomials \[ T(x)=T_0+\sum_{k\in K, k\le H}T_k\cos(2πkx), \qquad H>1, \] $T(x)\ge0$ for all $x$ and $T(0)=1$. Suppose that $0<h\le1/2$ and $K(h)$ is the class of functions \[ f(x)=\sum_{n=0}^{\infty}a_n\cos(2πnx) \] satisfying the following conditions: $a_n\ge0$ for all $n$, $f(0)=1$ and $f(x)=0$ for $h\le|x|\le1/2$. We consider an relation between extremum problem \[ δ(K)=\inf_{T\in\mathbf T(K)}T_0 \] and Turán extremum problem \[ A(h)=\sup_{f\in K(h)}a_0=\sup_{f\in K(h)}\int_{-h}^hf(x) dx \] for rational numbers $h=p/q$ a…
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 main purpose of this master thesis is to study the $LS$-sequences of points introduced by Carbone in \cite{Carbone} and find two generalizations of them to the unit square. Here we also present a new algorithm proposed by the same author in \cite{Carbone2} and we implement it in order to have a graphical description of these sequences. Chapter 1 includes a collection of results concerning the uniform ditribution theory and the discrepancy (we refer to \cite{Drmota_Tichy} and \cite{Kuipers_Niederreiter} for a complete survey on the matter). In Chapter 2 we focuse our attention on the $LS$-s…
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…
Although a generic uniformly distributed sequence has Poissonian pair correlations, only one explicit example has been found up to now. Additionally, it is even known that many classes of uniformly distributed sequences, like van der Corput sequences, Kronecker sequences and LS sequences, do not have Poissonian pair correlations. In this paper, we show that van der Corput sequences and the Kronecker sequence for the golden mean are as close to having Poissonian pair correlations as possible: they both have $α$-pair correlations for all $0 < α< 1$ but not for $α= 1$ which corresponds to Poisson…
By using the $q$-analogue of van der Corput's method we study the divisor function in an arithmetic progression to modulus $q$. We show that the expected asymptotic formula holds for a larger range of $q$ than was previously known, provided that $q$ has a certain factorisation.
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}.$$
In the present paper we study the asymptotic behavior of trigonometric products of the form $\prod_{k=1}^N 2 \sin(πx_k)$ for $N \to \infty$, where the numbers $ω=(x_k)_{k=1}^N$ are evenly distributed in the unit interval $[0,1]$. The main result are matching lower and upper bounds for such products in terms of the star-discrepancy of the underlying points $ω$, thereby improving earlier results obtained by Hlawka in 1969. Furthermore, we consider the special cases when the points $ω$ are the initial segment of a Kronecker or van der Corput sequence. The paper concludes with some probabilistic a…
The class of $(t,m,s)$-nets and $(t,s)$-sequences, introduced in their most general form by Niederreiter, are important examples of point sets and sequences that are commonly used in quasi-Monte Carlo algorithms for integration and approximation. Low-dimensional versions of $(t,m,s)$-nets and $(t,s)$-sequences, such as Hammersley point sets and van der Corput sequences, form important sub-classes, as they are interesting mathematical objects from a theoretical point of view, and simultaneously serve as examples that make it easier to understand the structural properties of $(t,m,s)$-nets and $…
In this paper, we present k sequences of Generalized Van der Laan Polynomials and Generalized Perrin Polynomials using Genaralized Fibonacci and Lucas Polynomials. We give some properties of these polynomials. We also obtain generalized order-k Van der Laan Numbers, k sequences of generalized order-k Van der Laan Numbers, generalized order-k Perrin Numbers and k sequences of generalized order-k Perrin Numbers. In addition, we examine the relationship between them.
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.
