arXiv subjects

Christian Weiß

Publications and source records attributed to Christian Weiß.

At least 19 recordsLinked to original sources

arXiv2026

Dimensions of Orbit Closures and Discrepancy for Dynamical $p$-adic Sequences

Classical discrepancy quantifies the irregularity of the distribution of a sequence in the unit interval. In this paper, we study the analogous notion for sequences in the ring of $p$-adic integers with a focus on the dynamically generated sequences. We prove that the orbits of ergodic $1$-Lipschitz self-maps of $\mathbb{Z}_p^d$ attain the optimal order of discrepancy and hence form low-discrepancy sequences. We also obtain bounds on the growth of the size of orbits of polynomial self-maps of $f: \mathbb{Z}_p^d \to \mathbb{Z}_p^d$ modulo $p^n$ for $d>1$. As a consequence, we show that orbit closures of $f$ have box dimension either zero or one. Our approach relies on the introduction of strong fixed points for such maps, together with several decomposition results for matrices over $\mathbb{Z}_p$.

math.NT↗

arXiv2026

An Explicit non-Poissonian Pair Correlation Function

A generic uniformly distributed random sequence on the unit interval has Poissonian pair correlations. Usually, the pair correlations statistic is therefore studied for equidistributed sequences. At the same time, there are only very few explicitly known examples of sequences with this property and many types of deterministic sequences have been proven to fail having the Poissonian pair correlation property. In this paper we study the pair correlation statistic in the non-uniform case and analyze the first elementary example of such a sequence, namely $x_n := \left\{ \frac{\log(2n-1)}{\log(2)} \right\}$, which is a standard low-dispersion sequence. The proof heavily relies on a full understanding of the gap structure of $(x_n)_{n=1}^N$. Furthermore, we discuss differences to the weak pair correlation function which turns out to be linear.

math.NT↗

arXiv2026

Linear Bounds for Differentiable Limits of Weak Pair Correlation Functions

For $s \geq 0$ and a parameter $0 < β< 1$, the weak pair correlation function $f_{N,β}(s)$ for the first $N \in \mathbb{N}$ elements of a sequence $(x_n)_{n \in \mathbb{N}} \subset[0,1]$ is evidently non-decreasing in $s$. Moreover, it satisfies $\lim_{N \to \infty} f_{N,β}(0) = 0$ if the elements of $(x_n)_{n \in \mathbb{N}}$ are distinct. Beyond these basic observations, little is known in general about the behavior of the limiting function. In this note, we investigate the situation in which the limit $f_β(s)=\lim_{N\to\infty} f_{N,β}(s)$ exists for all $s\ge 0$ and is differentiable in a neighborhood of the origin. Under these assumptions, we establish the bounds $2s \le f_β(s) \le f'_β(0)\, s,$ thereby providing general constraints on the limiting function.

math.NT↗

arXiv2026

Hammersley Point Sets and Inverse of Star-Discrepancy

We establish the existence of $N$-point sets in dimension $d$ whose star-discrepancy is bounded above by $2.4631832 \sqrt{\frac{d}{N}}$, where the numerical constant improves upon all previously known bounds. This improvement is obtained by combining a recent result by Gnewuch on bracketing numbers in high dimensions with discrepancy bounds for Hammersley point sets due to Atanassov in dimensions $1 \leq d \leq 4$.

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arXiv2025

Poissonian pair correlations for dependent random variables

We consider Poissonian pair correlations (PPC) for uniformly distributed sequences of random numbers with a dependency structure. More specifically, we treat two classes of dependent random variables which have widely been studied in the literature, namely sequences of jittered samples and random walks on the torus. We show that for the former class, the PPC property depends on how the finite sample is extended to an infinite sequence. Moreover, we prove that, under some mild assumptions, the random walk on the torus generically has PPC.

math.NT↗

arXiv2025

The Joint Asymptotic Distribution of Entropy and Complexity

We derive the asymptotic distribution of ordinal-pattern frequencies under weak dependence conditions and investigate the long-run covariance matrix not only analytically for moving-average, Gaussian, and the novel generalized coin-tossing processes, but also approximately by a simulation-based approach. Then, we deduce the asymptotic distribution of the entropy-complexity pair, which emerged as a popular tool for summarizing the time-series dynamics. Here, we make the necessary distinction between a uniform and a non-uniform ordinal pattern distribution and, thus, obtain two different limit theorems. On this basis, we consider a test for serial dependence and check its finite-sample performance. Moreover, we use our asymptotic results to approximate the estimation uncertainty of entropy-complexity pairs.

math.ST↗

arXiv2025

Weak Poissonian box correlations of higher order

Poissonian pair correlations have sparked interest within the mathematical community, because of their number theoretic properties, and their connections to quantum physics and probability theory, particularly uniformly distributed random numbers. Rather recently, several generalizations of the concept have been introduced, including weak Poissonian pair correlations and $k$-th order Poissonian correlations. In this paper, we propose a new generalized concept called $(k,m,β)$-Poissonian box correlations. We study their properties and more specifically their relation to uniform distribution theory, discrepancy theory, random numbers and gap distributions.

math.NT↗

arXiv2025

On the Pair Correlation Statistic of Sequences with Finite Gap Property

The limiting function $f(s)$ of the pair correlation \[ \frac{1}{N} \# \left\{ 1 \leq i\neq j\leq N \middle\vert \left\lVert x_i - x_j \right\rVert \leq \frac{s}{N} \right\} \] for a sequence $(x_N)_{N \in \mathbb{N}}$ on the torus $\mathbb{T}^1$ is said to be Poissonian if it exists and equals $2s$ for all $s \geq 0$. For instance, independent, uniformly distributed random variables generically have this property. Obviously $f(s)$ is always a monotonic function if existent. There are only few examples of sequences where $f(s) \neq 2s$, but where the limit can still be explicitly calculated. Therefore, it is an open question which types of functions $f(s)$ can or cannot appear here. In this note, we give a partial answer on this question by addressing the case that the number of different gap lengths in the sequence is finite and showing that $f$ cannot be continuous then.

math.NT↗

arXiv2024

Online Algorithms to Schedule a Proportionate Flexible Flow Shop of Batching Machines

This paper is the first to consider online algorithms to schedule a proportionate flexible flow shop of batching machines (PFFB). The scheduling model is motivated by manufacturing processes of individualized medicaments, which are used in modern medicine to treat some serious illnesses. We provide two different online algorithms, proving also lower bounds for the offline problem to compute their competitive ratios. The first algorithm is an easy-to-implement, general local scheduling heuristic. It is 2-competitive for PFFBs with an arbitrary number of stages and for several natural scheduling objectives. We also show that for total/average flow time, no deterministic algorithm with better competitive ratio exists. For the special case with two stages and the makespan or total completion time objective, we describe an improved algorithm that achieves the best possible competitive ratio $φ=\frac{1+\sqrt{5}}{2}$, the golden ratio. All our results also hold for proportionate (non-flexible) flow shops of batching machines (PFB) for which this is also the first paper to study online algorithms.

cs.DS↗

arXiv2024

P-adic Poissonian Pair Correlations via the Monna Map

Although the existence of sequences in the p-adic integers with Poissonian pair correlations has already been shown, no explicit examples had been found so far. In this note we discuss how to transfer real sequences with Poissonian pair correlations to the p-adic setting by making use of the Monna map.

math.NT↗

arXiv2024

Polynomial p-adic Low-Discrepancy Sequences

The classic example of a low-discrepancy sequence in $\mathbb{Z}_p$ is $(x_n) = an+b$ with $a \in \mathbb{Z}_p^x$ and $b \in \mathbb{Z}_p$. Here we address the non-linear case and show that a polynomial $f$ generates a low-discrepancy sequence in $\mathbb{Z}_p$ if and only if it is a permutation polynomial $\mod p$ and $\mod p^2$. By this it is possible to construct non-linear examples of low-discrepancy sequences in $\mathbb{Z}_p$ for all primes $p$. Moreover, we prove a criterion which decides for any given polynomial in $\mathbb{Z}_p$ with $p \in \left\{ 3,5, 7\right\}$ if it generates a low-discrepancy sequence. We also discuss connections to the theories of Poissonian pair correlations and real discrepancy.

math.NT↗

arXiv2023

On the finite pair correlation function of van der Corput sequences

In this note we derive an explicit formula for the finite empiric pair correlation function $F_N(s)$ of the van der Corput sequence in base $2$ for all $N \in \mathbb{N}$ and $s \geq 0$. The formula can be evaluated without explicit knowledge about the elements of the van der Corput sequence. Moreover, it can be immediately read off that $\lim_{N \to \infty} F_N(s)$ exists only for $0 \leq s \leq 1/2$.

math.NT↗

arXiv2023

New Bounds for the Extreme and the Star Discrepancy of Double-Infinite Matrices

According to Aistleitner and Weimar, there exist two-dimensional (double) infinite matrices whose star-discrepancy $D_N^{*s}$ of the first $N$ rows and $s$ columns, interpreted as $N$ points in $[0,1]^s$, satisfies an inequality of the form $$D_N^{*s} \leq \sqrtα \sqrt{A+B\frac{\ln(\log_2(N))}{s}}\sqrt{\frac{s}{N}}$$ with $α= ζ^{-1}(2) \approx 1.73, A=1165$ and $B=178$. These matrices are obtained by using i.i.d sequences, and the parameters $s$ and $N$ refer to the dimension and the sample size respectively. In this paper, we improve their result in two directions: First, we change the character of the equation so that the constant $A$ gets replaced by a value $A_s$ dependent on the dimension $s$ such that for $s>1$ we have $A_s<A$. Second, we generalize the result to the case of the (extreme) discrepancy. The paper is complemented by a section where we show numerical results for the dependence of the parameter $A_s$ on $s$.

math.NT↗

arXiv2023

The Pair Correlation Function of Multi-Dimensional Low-Discrepancy Sequences with Small Stochastic Error Terms

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

math.NT↗

arXiv2022

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 $α$-pair correlations with $0 < α< 1$. 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 $α$-pair correlations for $0 < α<1$. 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 $α$-pair correlations for $0 < α< 1$.

math.NT↗

arXiv2022

Approximation of Discrete Measures by Finite Point Sets

For a probability measure $μ$ on $[0,1]$ without discrete component, the best possible order of approximation by a finite point set in terms of the star-discrepancy is $\frac{1}{2N}$ as has been proven relatively recently. However, if $μ$ contains a discrete component no non-trivial lower bound holds in general because it is straightforward to construct examples without any approximation error in this case. This might explain, why the approximation of discrete measures on $[0,1]$ by finite point sets has so far not been completely covered in the existing literature. In this note, we close this gap by giving a complete description of the discrete case. Most importantly, we prove that for any discrete measure the best possible order of approximation is for infinitely many $N$ bounded from below by $\frac{1}{cN}$ for some constant $c \geq 2$ which depends on the measure. This implies, that for a finitely supported discrete measure on $[0,1]^d$ the known possible order of approximation $\frac{1}{N}$ is indeed the optimal one.

math.NT↗

arXiv2022

Neural calibration of hidden inhomogeneous Markov chains -- Information decompression in life insurance

Markov chains play a key role in a vast number of areas, including life insurance mathematics. Standard actuarial quantities as the premium value can be interpreted as compressed, lossy information about the underlying Markov process. We introduce a method to reconstruct the underlying Markov chain given collective information of a portfolio of contracts. Our neural architecture explainably characterizes the process by explicitly providing one-step transition probabilities. Further, we provide an intrinsic, economic model validation to inspect the quality of the information decompression. Lastly, our methodology is successfully tested for a realistic data set of German term life insurance contracts.

cs.LG↗

arXiv2021

Remarks on the Pair Correlation Statistic of Kronecker Sequences and Lattice Point Counting

In 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 $β$-pair correlations for all $0 < β< 1$.

math.NT↗