Van Der Corput Research Articles

VAN DER CORPUT LEMMA WITH BESSEL FUNCTIONS

In this article, we study analogues of the van der Corput lemmas [19] involving Bessel functions. In harmonic analysis, one of the most important estimates is the van der Corput lemma, which is an estimate of the oscillatory integrals. This estimate was first obtained by the Dutch mathematician Johannes Gaultherus van der Corput. Van der Corput interested in the behavior for large positive λ of the oscillatory integral R b a e iλφ(x)ψ(x)dx, where φ is a real-valued smooth function (the phase) and ψ is complex valued smooth function (amplitude). In case a = −∞, b = +∞, it is assumed that ψ has a compact support in R. In our case we replace the exponential function with the Bessel functions, to study oscillatory integrals appearing in the analysis of wave equation with singular damping. More specifically, we study integral of the form I(λ) = R b a Jn(λφ(x))ψ(x)dx for the range n = 0, where ψ ∈ C and smooth, and λ is a positive real number that can vary.The generalisations of the van der Corput lemma is proved. As an application of the above results, the generalised Riemann-Lebesgue lemma is considered.

A Generalized Polar-Coordinate Integration Formula with Applications to the Study of Convolution Powers of Complex-Valued Functions on $${\mathbb {Z}}^d$$

In this article, we consider a class of functions on $${\mathbb {R}}^d$$ , called positive homogeneous functions, which interact well with certain continuous one-parameter groups of (generally anisotropic) dilations. Generalizing the Euclidean norm, positive homogeneous functions appear naturally in the study of convolution powers of complex-valued functions on $${\mathbb {Z}}^d$$ . As the spherical measure is a Radon measure on the unit sphere which is invariant under the symmetry group of the Euclidean norm, to each positive homogeneous function P, we construct a Radon measure $$\sigma _P$$ on $$S=\{\eta \in {\mathbb {R}}^d:P(\eta )=1\}$$ which is invariant under the symmetry group of P. With this measure, we prove a generalization of the classical polar-coordinate integration formula and deduce a number of corollaries in this setting. We then turn to the study of convolution powers of complex functions on $${\mathbb {Z}}^d$$ and certain oscillatory integrals which arise naturally in that context. Armed with our integration formula and the Van der Corput lemma, we establish sup-norm-type estimates for convolution powers; this result is new and partially extends results of Randles and Saloff-Coste (J Fourier Anal Appl 21(4):754–798, 2015; Rev Mat Iberoam 33(3):1045–1121, 2017).

Arithmetic exponent pairs for algebraic trace functions and applications

We study short sums of algebraic trace functions via the $q$-analogue of van der Corput method, and develop methods of arithmetic exponent pairs that coincide with the classical case while the moduli has sufficiently good factorizations. As an application, we prove a quadratic analogue of Brun-Titchmarsh theorem on average, bounding the number of primes $p\leqslant X$ with $p^2+1\equiv0\pmod q$. The other two applications include a larger level of distribution of divisor functions in arithmetic progressions and a sub-Weyl subconvex bound of Dirichlet $L$-functions studied previously by Irving.

Van der Corput inequality for real line and Wiener-Wintner theorem for amenable groups

We extend the classical van der Corput inequality to the real line. As a consequence, we obtain a simple proof of the Wiener-Wintner theorem for the RR-action which assert that for any family of maps (Tt)t∈R(Tt)t∈R acting on the Lebesgue measure space (Ω,A,μ)(Ω,A,μ), where μμ is a probability measure and for any t∈Rt∈R, TtTt is measure-preserving transformation on measure space (Ω,A,μ)(Ω,A,μ) with Tt∘Ts=Tt+sTt∘Ts=Tt+s, for any t,s∈Rt,s∈R. Then, for any f∈L1(μ)f∈L1(μ), there is a single null set off which $\displaystyle \lim_{T \rightarrow +\infty} \frac{1}{T}\int_{0}^{T} f(T_t\omega) e^{2 i \pi \theta t} dt$limT→+∞1T∫0Tf(Ttω)e2iπθtdt exists for all θ∈θ∈\RRR. We further present the joining proof of the amenable group version of Wiener-Wintner theorem due to Ornstein and Weiss.

Sequences with almost Poissonian pair correlations

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 Poissonian pair correlations.

Van der Corput lemmas for Mittag-Leffler functions. II. α–directions

The paper is devoted to study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory integrals appearing in the analysis of time-fractional partial differential equations. More specifically, we study integral of the form Iα,β(λ)=∫REα,β(iαλϕ(x))ψ(x)dx, for the range 0<α≤2,β>0. This extends the variety of estimates obtained in the first part, where integrals with functions Eα,β(iλϕ(x)) have been studied. Several generalisations of the van der Corput lemmas are proved. As an application of the above results, the generalised Riemann-Lebesgue lemma, the Cauchy problem for the time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

Uniform Distribution of the Weighted Sum-of-Digits Functions

Abstract The higher-dimensional generalization of the weighted q-adic sum-of-digits functions sq,γ (n), n =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., d-dimensional van der Corput-Halton or d-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted q-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function g(x)= x implies the uniform distribution modulo one of the weighted q-adic sum-of-digits function sq,γ (n), n = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences h 1 sq, γ (n)+h 2 sq,γ (n +1), where h 1 and h 2 are integers such that h 1 + h 2 ≠ 0 and that the akin two-dimensional sequence sq,γ (n), sq,γ (n +1) cannot be uniformly distributed modulo one if q ≥ 3. The properties of the two-dimensional sequence sq,γ (n),s q,γ (n +1), n =0, 1, 2,..., will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.

Remarks on continuously distributed sequences

In the first part of the paper we define the notion of the density as certain type of finitely additive probability measure and the distribution function of sequences with respect to the density. Then we derive some simple criterions providing the continuity of the distribution function of given sequence. These criterions we apply to the van der Corput's sequences. The Weyl's type criterions of continuity of the distribution function are proven.

The Weyl formula for planar annuli

We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the Dirichlet Laplacian associated with a planar annulus to a lattice point counting problem associated with a special domain in R2. Unlike other lattice point problems, the one arisen naturally here has interesting features that lattice points under consideration are translated by various amounts and the curvature of the boundary is unbounded. By transforming this problem into a relatively standard form and using classical van der Corput's bounds, we obtain a two-term Weyl formula for the eigenvalue counting function for the planar annulus with a remainder of size O(μ2/3). If we additionally assume that certain tangent has rational slope, we obtain an improved remainder estimate of the same strength as Huxley's bound in the Gauss circle problem, namely O(μ131/208(log⁡μ)18627/8320). As a by-product of our lattice point counting results, we readily obtain this Huxley-type remainder estimate in the two-term Weyl formula for planar disks.

Lax-Wendroff methods on highly non-uniform meshes. Dedicated to the Memory of Blair Swartz (1932–2019)

The standard Lax-Wendroff scheme with the conservative Lax-Friedrichs nodal predictor on highly non-uniform meshes produces serious oscillations, making it useless on such meshes. Wendroff and White (1989) [13] proposed two versions (WW and WWJp) with different predictors which work robustly on such meshes. Both WW and WWJp are second order accurate. We investigate how these methods behave on highly non-uniform meshes of three types (Pike, cluster and van der Corput) for 1D smooth solutions of the Burgers and the Euler equations. The WW and WWJp methods are extended to 2D and tested on smooth solutions of the Euler equations on 2D meshes created by the Cartesian product of 1D highly non-uniform meshes. We have not been able to find any significant difference between the WW and WWJp results, thus the simpler WW should be preferred.

Convergence Rates on Spectral Orthogonal Projection Approximation for Functions of Algebraic and Logarithmatic Regularities

Based on the Hilb type formula between Jacobi polynomials and Bessel functions, optimal decay rates on the Jacobi expansion coefficients are derived by applying van der Corput type lemmas for funct...

Squarefree Integers in Arithmetic Progressions to Smooth Moduli

Abstract Let $\varepsilon&gt; 0$ be sufficiently small and let $0 &lt; \eta &lt; 1/522$ . We show that if X is large enough in terms of $\varepsilon $ , then for any squarefree integer $q \leq X^{196/261-\varepsilon }$ that is $X^{\eta }$ -smooth one can obtain an asymptotic formula with power-saving error term for the number of squarefree integers in an arithmetic progression $a \pmod {q}$ , with $(a,q) = 1$ . In the case of squarefree, smooth moduli this improves upon previous work of Nunes, in which $196/261 = 0.75096\ldots $ was replaced by $25/36 = 0.69\overline {4}$ . This also establishes a level of distribution for a positive density set of moduli that improves upon a result of Hooley. We show more generally that one can break the $X^{3/4}$ -barrier for a density 1 set of $X^{\eta }$ -smooth moduli q (without the squarefree condition). Our proof appeals to the q-analogue of the van der Corput method of exponential sums, due to Heath-Brown, to reduce the task to estimating correlations of certain Kloosterman-type complete exponential sums modulo prime powers. In the prime case we obtain a power-saving bound via a cohomological treatment of these complete sums, while in the higher prime power case we establish savings of this kind using p-adic methods.

Greedy energy minimization can count in binary: point charges and the van der Corput sequence

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 (1) symmetric f(x) = f(1-x), (2) twice differentiable on (0, 1), and (3) 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 xN=argminx∈[0,1)∑k=0N-1f(|x-xk|).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} x_{N} = \\arg \\min _{x \\in [0,1)} \\sum _{k=0}^{N-1}{f(|x-x_k|)}. \\end{aligned}$$\\end{document}We prove that if we start this construction with the single element x_0=0, then all arising constructions are permutations of the van der Corput sequence (counting in binary and reflected about the comma): greedy energy minimization recovers the way we count in binary. This gives a new construction of the classical van der Corput sequence. The special case f(x) = 1-log (2 sin (pi x)) answers a question of Steinerberger. Interestingly, the point sets we derive are also known in a different context as Leja sequences on the unit disk.

An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications

Abstract Generalized polynomials are mappings obtained from the conventional polynomials by the use of the operations of addition and multiplication and taking the integer part. Extending the classical theorem of Weyl on equidistribution of polynomials, we show that a generalized polynomial $q(n)$ has the property that the sequence $(q(n) \lambda )_{n \in \mathbf{Z}}$ is well-distributed $\bmod \, 1$ for all but countably many $\lambda \in{\mathbf R}$ if and only if $\lim\nolimits _{\substack{|n| \rightarrow \infty \\ n \notin J}} |q(n)| = \infty $ for some (possibly empty) set $J$ having zero natural density in $\mathbf{Z}$. We also prove a version of this theorem along the primes (which may be viewed as an extension of classical results of Vinogradov and Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets.

Exponential improvements for superball packing upper bounds

We prove that for all fixed p>2, the translative packing density of unit ℓp-balls in Rn is at most 2(γp+o(1))n with γp<−1/p. This is the first exponential improvement in high dimensions since van der Corput and Schaake (1936).

Influence of Initializing Krill Herd Algorithm With Low-Discrepancy Sequences

The krill herd (KH) algorithm is a global metaheuristic algorithm that was initially proposed for solving continuous optimization problems. The KH algorithm, since inception, has generated considerable real-world application interests in the research community. The standard algorithm solution implementation steps follow the initialization mechanism, which relies mainly on educated guesses or random initialization solution generation. Therefore, to improve the performance of the KH algorithm, the current study is set to investigate the influence of initializing the KH algorithm with three low-discrepancy sequences, such as the Faure sequence, Sobol sequence, and Van der Corput sequence. These low-discrepancy sequences are known to be more uniformly distributed across the problem search space than the commonly used random number initialization method. The study also evaluates the influence of population size on the performance of the proposed variants of the improved KH algorithms. The experimental results show significant improvements for the enhanced KH algorithms in terms of performance and the quality of solutions obtained; particularly on standard benchmarked high-dimension test problem instances, where the enhanced KH variants outperformed the existing basic KH algorithm for all the test functions evaluated. Similarly, the results for low dimension test cases showed less sensitivity to the initialization schemes, as the performance of our proposed improved scheme was comparable to that of the basic KH algorithm. However, in most cases, as the problem dimension was scaled up, the enhanced KH outperformed the basic KH. Evaluation results based on the population size of the algorithm, revealed that when the number of Krill is set at 25, the Sobol based KH initialization scheme performed better than did the other methods. Although, the Van der Corput and Faure based KH initialization schemes showed similar sensitivity when the dimension was set at 20. As we varied the population size of Krill, it was observed that the performance of the Sobol based KH initialization scheme deteriorated, whereas the other two methods showed superior performance. Overall, the findings from this study revealed that there are significant improvements in the performance of KH algorithm when initialized with low-discrepancy sequences.

The numerical oscillation suppression method for the large time step scheme

The large time step scheme, which is commonly used in hydrodynamic numerical simulation, adopts the multielement reconstruction method in order to improve both the computational efficiency and accuracy; however, this results in numerical oscillation and calculation divergence. Therefore, it is necessary to develop methods to suppress the occurrence of numerical oscillation near the discontinuities. The numerical oscillation that is induced can be suppressed by using the random choice method (RCM) with the van der Corput random number sequence. In order to optimize the suppression performance, error analysis has been carried out in this paper for the calculation results from the RCM with the popular van der Corput random sequence combinations for selecting the optimal sequence combination. In order to further eliminate any short-wave numerical oscillations near the discontinuities, a method that combines the RCM with a nonlinear filter, which often used in aeroacoustic calculations, has been proposed in this paper to suppress short-wave oscillations. Specifically, the RCM and the nonlinear filter method were sequentially implemented in order to process the calculation results at each time step. The results showed that this method had a good suppression effect on the numerical oscillations near the discontinuities, which was demonstrated by the numerical experiments.

Some negative results related to Poissonian pair correlation problems

We say that a sequence (xn)n∈N in [0,1) has Poissonian pair correlations if limN→∞1N#1≤l≠m≤N:‖xl−xm‖≤sN=2sfor every s≥0. The aim of this article is twofold. First, we will establish a gap theorem which allows to deduce that a sequence (xn)n∈N of real numbers in [0,1) having a certain weak gap structure, cannot have Poissonian pair correlations. This result covers a broad class of sequences, e.g., Kronecker sequences, the van der Corput sequence and more generally LS-sequences of points and digital (t,1)-sequences. Additionally, this theorem enables us to derive negative pair correlation results for sequences of the form ({anα})n∈N, where (an)n∈N is a strictly increasing sequence of integers with maximal order of additive energy, a notion that plays an important role in many fields, e.g., additive combinatorics, and is strongly connected to Poissonian pair correlation problems. These statements are not only metrical results, but hold for all possible choices of α.Second, in this note we study the pair correlation statistic for sequences of the form, xn={bnα},n=1,2,3,…, with an integer b≥2, where we choose α as the Stoneham number or as an infinite de Bruijn word. We will prove that both instances fail to have the Poissonian property.

APPROXIMATE FUNCTIONAL EQUATION AND UPPER BOUNDS FOR THE BARNES DOUBLE ZETA-FUNCTION

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 $ \zeta_2 (s, \alpha ; v, w ) = \sum_{m=0}^\infty \sum_{n=0}^\infty (\alpha+vm+wn)^{-s} $. Also, applying this approximate functional equation and the van der Corput method, we obtain upper bounds for $ \zeta_2(1/2 + it, \alpha ; v, w) $ and $ \zeta_2(3/2 + it, \alpha ; v, w) $ with respect to $ t $ as $ t \rightarrow \infty $.

On the Intriguing Search for Good Permutations

Abstract 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 versus using a sequence of permutations. Throughout, we collect, sharpen and extend open questions which all stem from the extensive work of Henri and his coworkers and which will hopefully inspire more work in the future.