Kronecker Sequences Research Articles

One-dimensional quasi-uniform Kronecker sequences

In this short note, we prove that the one-dimensional Kronecker sequence iαmod1,i=0,1,2,…,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i\\alpha \\bmod 1, i=0,1,2,\\ldots ,$$\\end{document} is quasi-uniform over the unit interval [0, 1] if and only if α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} is a badly approximable number. Our elementary proof relies on a result on the three-gap theorem for Kronecker sequences due to Halton (Proc Camb Philos Soc, 61:665–670, 1965).

Kronecker Sequences with Many Distances

The three gap theorem states that for any α ∈ R and N ∈ N , the number of different gaps between consecutive nα ( mod 1 ) for n ∈ { 1 , … , N } is at most 3. Biringer and Schmidt (2008) instead considered the distance from each point to its nearest neighbor, generalizing to higher dimensions. We denote the maximum number of distances in T d using the p-norm by g ¯ p d so that g ¯ p 1 = 3 . Haynes and Marklof (2021) showed that each example with arbitrary α and N gives a generic lower bound, and that g ¯ 2 2 = 5 and g ¯ 2 d ≤ σ d + 1 where σd is the kissing number. They gave an example showing g ¯ 2 3 ≥ 7 . Our examples that show g ¯ 2 3 ≥ 9 and also g ¯ 2 4 ≥ 11 , g ¯ 2 5 ≥ 13 and g ¯ 2 6 ≥ 14 . Haynes and Ramirez (2021) showed that g ¯ ∞ d ≤ 2 d + 1 and that this is sharp for d ≤ 3 . We provide a numerical example to show g ¯ ∞ 4 ≥ 15 , and a proof that g ¯ ∞ d ≥ 2 d − 1 + 1 in general. Results for p = ∞ and σd imply that g ¯ p d depends on p for d ≥ 11 and we conjecture this for d ≥ 4 . For d ≤ 3 we expect that g ¯ p d = { 3 , 5 , 9 } for d = { 1 , 2 , 3 } respectively, independent of p. For d = 1 this is trivial, for d = 2 we show that g ¯ p 2 ≥ 5 and for d = 3 we provide numerical examples suggesting that g ¯ p 3 ≥ 9 .

The pair correlation function of multi-dimensional low-discrepancy sequences with small stochastic error terms

In any dimension d≥2, there is no known example of a low-discrepancy sequence which possesses Poisssonian pair correlations. This is in some sense rather surprising, because low-discrepancy sequences always have β-Poissonian pair correlations for all 0<β<1d and are therefore arbitrarily close to having Poissonian pair correlations (which corresponds to the case β=1d). In this paper, we further elaborate on the closeness of the two notions. We show that d-dimensional Kronecker sequences for badly approximable vectors α→ with an arbitrary small uniformly distributed stochastic error term generically have β=1d-Poissonian pair correlations.

Upper bound on saturation time of metric graphs by intervals moving on them

In this paper we study a dynamical system of intervals moving on a metric graph with incommensurable edge lengths. This system can be generally viewed as a collection of congruent intervals moving around the network at unit speed, propagating along all respective incident edges whenever they pass through a vertex. In particular, we analyse the phenomenon of saturation: a state when the entire graph is covered by the moving intervals. Our main contributions are as follows: (1) we prove the existence of the finite moment of permanent saturation for any metric graph with incommensurable edge lengths and any positive length of the intervals; (2) we present an upper bound on the moment of permanent saturation. To show the validity of our results, we reduce the system of moving intervals to the system of dispersing moving points, for the analysis of which we mainly use methods from discrepancy theory and number theory, in particular Kronecker sequences and the famous Three Gap Theorem.

Remarks on the pair correlation statistic of Kronecker sequences and lattice point counting

AbstractIn this short note, we reformulate the task of calculating the pair correlation statistics of a Kronecker sequence as a lattice point counting problem. This can be done analogously to the lattice based approach which was used to (re-)prove the famous three gap property for Kronecker sequences. We show that recently developed lattice point counting techniques can then be applied to derive that a certain class of Kronecker sequences have $$\beta $$ β -pair correlations for all $$0&lt; \beta &lt; 1$$ 0 &lt; β &lt; 1 .

Speed of convergence of Weyl sums over Kronecker sequences

We study the speed of convergence in the numerical integration with Weyl sums over Kronecker sequences in the torus, 1N∑n=1Nfx+nα-∫Tdf(y)dy.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\dfrac{1}{N}\\sum _{n=1}^{N}f\\left( x+n\\alpha \\right) -\\int _{\\mathbb {T}^{d} }f(y)dy. \\end{aligned}$$\\end{document}

Some connections between discrepancy, finite gap properties, and pair correlations

A generic uniformly distributed sequence (x_n)_{n in mathbb {N}} in [0, 1) possesses Poissonian pair correlations (PPC). Vice versa, it has been proven that a sequence with PPC is uniformly distributed. Grepstad and Larcher gave an explicit upper bound for the discrepancy of a sequence given that it has PPC. As a first result, we generalize here their result to the case of alpha -pair correlations with 0< alpha < 1. Since the highest possible level of uniformity is achieved by low-discrepancy sequences it is tempting to assume that there are examples of such sequences which also have PPC. Although there are no such known examples, we prove that every low-discrepancy sequence has at least alpha -pair correlations for 0< alpha <1. According to Larcher and Stockinger, the reason why many known classes of low-discrepancy sequences fail to have PPC is their finite gap property. In this article, we furthermore show that the discrepancy of a sequence with the finite gap property plus a condition on the distribution of the different gap lengths can be estimated. As a concrete application of this estimation, we re-prove the fact that van der Corput and Kronecker sequences are low-discrepancy sequences. Consequently, it follows from the finite gap property that these sequences have alpha -pair correlations for 0< alpha < 1.

Multi-dimensional Kronecker sequences with a small number of gap lengths

Abstract Recently, generalizations of the classical Three Gap Theorem to higher dimensions attracted a lot of attention. In particular, upper bounds for the number of nearest neighbor distances have been established for the Euclidean and the maximum metric. It was proved that a generic multi-dimensional Kronecker attains the maximal possible number of different gap lengths for every sub-exponential subsequence. We mirror this result in dimension d ∈ {2, 3} by constructing Kronecker sequences which have a surprisingly low number of different nearest neighbor distances for infinitely N ∈ ℕ. Our proof relies on simple arguments from the theory of continued fractions.

A Five Distance Theorem for Kronecker Sequences

Abstract The three-distance theorem (also known as the three-gap theorem or Steinhaus problem) states that, for any given real number $\alpha $ and integer $N$, there are at most three values for the distances between consecutive elements of the Kronecker sequence $\alpha , 2\alpha ,\ldots , N\alpha $ mod 1. In this paper, we consider a natural generalization of the three-distance theorem to the higher-dimensional Kronecker sequence $\vec \alpha , 2\vec \alpha ,\ldots , N\vec \alpha $ modulo an integer lattice. We prove that in 2D, there are at most five values that can arise as a distance between nearest neighbors, for all choices of $\vec \alpha $ and $N$. Furthermore, for almost every $\vec \alpha $, five distinct distances indeed appear for infinitely many $N$ and hence five is the best possible general upper bound. In higher dimensions, we have similar explicit, but less precise, upper bounds. For instance, in 3D, our bound is 13, though we conjecture the truth to be 9. We furthermore study the number of possible distances from a point to its nearest neighbor in a restricted cone of directions. This may be viewed as a generalization of the gap length in 1D. For large cone angles, we use geometric arguments to produce explicit bounds directly analogous to the three-distance theorem. For small cone angles, we use ergodic theory of homogeneous flows in the space of unimodular lattices to show that the number of distinct lengths is (1) unbounded for almost all $\vec \alpha $ and (2) bounded for $\vec \alpha $ that satisfy certain Diophantine conditions.

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.

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.

On the Wasserstein distance between classical sequences and the Lebesgue measure

We discuss the classical problem of measuring the regularity of distribution of sets of $N$ points in $\mathbb{T}^d$. A recent line of investigation is to study the cost ($=$ mass $\times$ distance) necessary to move Dirac measures placed in these points to the uniform distribution. We show that Kronecker sequences satisfy optimal transport distance in $d \geq 3$ dimensions. This shows that for differentiable $f: \mathbb{T}^d \rightarrow \mathbb{R}$ and badly approximable vectors $\alpha \in \mathbb{R}^d$, we have $$ | \int_{\mathbb{T}^d} f(x) dx - \frac{1}{N} \sum_{k=1}^{N} f(k \alpha) | \leq c_{\alpha} \frac{ \| \nabla f\|^{(d-1)/d}_{L^{\infty}}\| \nabla f\|^{1/d}_{L^{2}} }{N^{1/d}}.$$ We note that the result is uniformly true for a sequence instead of a set. Simultaneously, it refines the classical integration error for Lipschitz functions, $\| \nabla f\|_{L^{\infty}} N^{-1/d}$. We obtain a similar improvement for numerical integration with respect to the regular grid. The main ingredient is an estimate involving Fourier coefficients of a measure; this allows for existing estimates to be conviently `recycled'. We present several open problems.

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.

On The Classification of Ls-Sequences

Abstract This paper addresses the question whether the LS-sequences constructed in [Car12] yield indeed a new family of low-discrepancy sequences. While it is well known that the case S = 0 corresponds to van der Corput sequences, we prove here that the case S = 1 can be traced back to symmetrized Kronecker sequences and moreover that for S ≥ 2 none of these two types occurs anymore. In addition, our approach allows for an improved discrepancy bound for S = 1 and L arbitrary.

Metrical Star Discrepancy Bounds for Lacunary Subsequences of Digital Kronecker-Sequences and Polynomial Tractability

Abstract The star discrepancy D N * ( 𝒫 ) $D_N^* \left( {\cal P} \right)$ is a quantitative measure for the irregularity of distribution of a finite point set 𝒫 in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer N ≥ 2 there are point sets 𝒫 in [0, 1) d with |𝒫| = N and D N * ( 𝒫 ) = O ( ( log N ) d - 1 / N ) $D_N^* \left( {\cal P} \right) = O\left( {\left( {\log \,N} \right)^{d - 1} /N} \right)$ . However, for small N compared to the dimension d this asymptotically excellent bound is useless (e.g., for N ≤ ed −1). In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Woźniakowski that for every integer N ≥ 2there exist point sets 𝒫 in [0, 1) d with |𝒫| = N and D N * ( 𝒫 ) ≤ C d / N $D_N^* \left( {\cal P} \right) \le C\sqrt {d/N}$ . Although not optimal in an asymptotic sense in N, this upper bound has a much better (and even optimal) dependence on the dimension d. Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties is known. Quite recently Löbbe studied lacunary subsequences of Kronecker’s (n α)-sequence and showed a metrical discrepancy bound of the form C d ( log d ) / N $C\sqrt {d\left({\log \,d} \right)/N}$ with implied absolute constant C&gt; 0 independent of N and d. In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.

Kronecker–Halton sequences in [formula omitted

In this paper we investigate the distribution properties of hybrid sequences which are made by combining Halton sequences in the ring of polynomials and digital Kronecker sequences. We give a full criterion for the uniform distribution and prove results on the discrepancy of such hybrid sequences.

Lattice-based integration algorithms: Kronecker sequences and rank-1 lattices

We prove upper bounds on the order of convergence of lattice-based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov spaces of dominating mixed smoothness mathring{mathbf {B}}^s_{p,theta }, which also comprise the concept of Sobolev spaces of dominating mixed smoothness mathring{mathbf {H}}^s_{p} as special cases. The considered algorithms are quasi-Monte Carlo rules with underlying nodes from T_Nleft( mathbb {Z}^dright) cap [0,1)^d, where T_N is a real invertible generator matrix of size d. For such rules, the worst-case error can be bounded in terms of the Zaremba index of the lattice mathbb {X}_N=T_Nleft( mathbb {Z}^dright) . We apply this result to Kronecker lattices and to rank-1 lattice point sets, which both lead to optimal error bounds up to log N-factors for arbitrary smoothness s. The advantage of Kronecker lattices and classical lattice point sets is that the run-time of algorithms generating these point sets is very short.

On the discrepancy of two-dimensional perturbed Halton--Kronecker sequences and lacunary trigonometric products

We consider the star discrepancy of two-dimensional sequences made up as a hybrid between a Kronecker sequence and a perturbed Halton sequence in base 2, where the perturbation is achieved by a digital-sequence construction in the sense of Niederreiter wh

On evil Kronecker sequences and lacunary trigonometric products

On evil Kronecker sequences and lacunary trigonometric products

Modeling of non‐stationary vibration signals based on the modified Kronecker sequences

AbstractThe paper establishes a representation model for non‐stationary random vibration signals based on the modified Kronecker sequences. The modified Kronecker sequence constructed via generalizing golden ratio is one of the special types of low discrepancy sequences which have better dimensional projections [1]. The actual modeling and simulation of non‐stationary random data is more suitable for seismological signals and not for the vehicle vibrations [2, 3]. Under these circumstances, this paper presents a new algorithm for finding the modified Kronecker sequences in order to generate non‐stationary vehicle vibration signals which mostly withhold the amplitude‐frequency‐time distribution of the sample signal. (© 2015 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)