Related papers: Low Discrepancy Digital Kronecker-Van der Corput S…

Dick proved that all order $2$ digital nets satisfy optimal upper bounds of the $L_2$-discrepancy. We give an alternative proof for this fact using Haar bases. Furthermore, we prove that all digital nets satisfy optimal upper bounds of the…

Score-matching generative models have proven successful at sampling from complex high-dimensional data distributions. In many applications, this distribution is believed to concentrate on a much lower $d$-dimensional manifold embedded into…

The spherical cap discrepancy is a prominent measure of uniformity for sets on the d-dimensional sphere. It is particularly important for estimating the integration error for certain classes of functions on the sphere. Building on a…

An important result of H. Weyl states that for every sequence $\left(a_{n}\right)_{n \geq 1}$ of distinct positive integers the sequence of fractional parts of $\left(a_{n} \alpha \right)_{n\geq 1}$ is uniformly distributed modulo one for…

Consider two series $$\sum_{n=1}^\infty\frac{\sin^n\pi\theta n}{n^\alpha},\quad\sum_{n=1}^\infty\frac{\cos^n\pi\theta n}{n^\alpha}.$$ We show that number-theoretical properties of $\theta$ have a strong effect on the convergence when…

We randomly construct various subsets $\Lambda$ of the integers which have both smallness and largeness properties. They are small since they are very close, in various meanings, to Sidon sets: the continuous functions with spectrum in…

The identification of increasingly smaller signal from objects observed with a non-perfect instrument in a noisy environment poses a challenge for a statistically clean data analysis. We want to compute the probability of frequencies…

We provide the detailed asymptotic behavior for first-order aggregation models of heterogeneous oscillators. Due to the dissimilarity of natural frequencies, one could expect that all relative distances converge to definite positive value…

This paper studies the properties of a new lower bound for the natural pseudo-distance. The natural pseudo-distance is a dissimilarity measure between shapes, where a shape is viewed as a topological space endowed with a real-valued…

We prove a generalization of van der Corput's difference theorem for sequences of vectors in a Hilbert space. This generalization is obtained by establishing a connection between sequences of vectors in the first Hilbert space with a vector…

Low-discrepancy (LD) sequences have been extensively used as efficient experimental designs across many scientific disciplines. QMCPy (https://qmcsoftware.github.io/QMCSoftware/) is an accessible Python library which provides a unified…

Kernel discrepancies are a powerful tool for analyzing worst-case errors in quasi-Monte Carlo (QMC) methods. Building on recent advances in optimizing such discrepancy measures, we extend the subset selection problem to the setting of…

According to the Erd\H{o}s discrepancy conjecture, for any infinite $\pm 1$ sequence, there exists a homogeneous arithmetic progression of unbounded discrepancy. In other words, for any $\pm 1$ sequence $(x_1,x_2,...)$ and a discrepancy…

The approximate Carath\'eodory problem in general form is as follows: Given two symmetric convex bodies $P,Q \subseteq \mathbb{R}^m$, a parameter $k \in \mathbb{N}$ and $\mathbf{z} \in \textrm{conv}(X)$ with $X \subseteq P$, find…

For Lebesgue generic $(x_1,x_2)\in \mathbb{R}^2$, we investigate the distribution of small values of products $q\cdot \|qx_1\| \cdot \|qx_2\|$ with $q\in\mathbb{N}$, where $\|\cdot \|$ denotes the distance to the closest integer. The main…

A long-standing conjecture of Littlewood about simultaneous Diophantine approximation has an analogous problem for a field of formal Laurent series $\mathbb{F}(\!(t^{-1})\!)$. That is, we can ask whether for any series $\Theta$, $\Phi$ and…

In discrepancy minimization problems, we are given a family of sets $\mathcal{S} = \{S_1,\dots,S_m\}$, with each $S_i \in \mathcal{S}$ a subset of some universe $U = \{u_1,\dots,u_n\}$ of $n$ elements. The goal is to find a coloring $\chi :…

For data assumed to come from a finite mixture with an unknown number of components, it has become common to use Dirichlet process mixtures (DPMs) not only for density estimation, but also for inferences about the number of components. The…

Let $ (H_s(n))_{n \geq 1} $ be an $s-$dimensional Halton's sequence. Let $D_N$ be the discrepancy of the sequence $ (H_s(n))_{n = 1}^{N} $. It is known that $ND_N =O(\ln^s N)$ as $N \to \infty $. In this paper we prove that this estimate is…

We construct the Laurent phenomenon algebras the cluster variables of which satisfy the discrete BKP equation and other difference equations obtained by its reduction. These Laurent phenomenon algebras are constructed from seeds with a…