Related papers: On lower bounds for the L_2-discrepancy
We give an improved lower bound for the $L_2$-discrepancy of finite point sets in the unit square.
We find the best asymptotic lower bounds for the coefficient of the leading term of the $L_1$ norm of the two-dimensional (axis-parallel) discrepancy that can be obtained by K.Roth's orthogonal function method among a large class of test…
It is a well-known conjecture in the theory of irregularities of distribution that the L1 norm of the discrepancy function of an N-point set satisfies the same asymptotic lower bounds as its L^2 norm. In dimension d=2 this fact has been…
In this book chapter we survey known approaches and algorithms to compute discrepancy measures of point sets. After providing an introduction which puts the calculation of discrepancy measures in a more general context, we focus on the…
The irregularities of a distribution of $N$ points in the unit interval are often measured with various notions of discrepancy. The discrepancy function can be defined with respect to intervals of the form $[0,t)\subset [0,1)$ or arbitrary…
A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq 3. It follows from the average case bound of Klaus Roth that the L-infty norm of D_N is at…
Two popular and often applied methods to obtain two-dimensional point sets with the optimal order of $L_p$ discrepancy are digit scrambling and symmetrization. In this paper we combine these two techniques and symmetrize $b$-adic Hammersley…
We consider the problem of finding, for a given quadratic measure of non-uniformity of a set of $N$ points (such as $L_2$ star-discrepancy or diaphony), the asymptotic distribution of this discrepancy for truly random points in the limit…
In this paper we study the extreme and the periodic $L_2$ discrepancy of plane point sets. The extreme discrepancy is based on arbitrary rectangles as test sets whereas the periodic discrepancy uses "periodic intervals", which can be seen…
We initiate the study of quadratic discrepancy for finite point sets on the Heisenberg group $\mathbb H^n$ with respect to upper Ahlfors regular probability measures. For a natural family of test sets given by left translations and…
We use the Haar function system in order to study the $L_2$ discrepancy of a class of digital $(0,n,2)$-nets. Our approach yields exact formulas for this quantity, which measures the irregularities of distribution of a set of points in the…
We study the extreme and the periodic $L_p$ discrepancy of point sets in the $d$-dimensional unit cube. The extreme discrepancy uses arbitrary sub-intervals of the unit cube as test sets, whereas the periodic discrepancy is based on…
Let $(H(n))_{n \geq 0} $ be a $2-$dimensional Halton's sequence. Let $D_{2} ( (H(n))_{n=0}^{N-1}) $ be the $L_2$-discrepancy of $ (H_n)_{n=0}^{N-1} $. It is known that $\limsup_{N \to \infty } (\log N)^{-1} D_{2} ( H(n) )_{n=0}^{N-1} >0$.…
In this article we survey recent results on the explicit construction of finite point sets and infinite sequences with optimal order of $\mathcal{L}_q$ discrepancy. In 1954 Roth proved a lower bound for the $\mathcal{L}_2$ discrepancy of…
We investigate $L_2$-discrepancies of what we call weak Latin hypercubes. In this case it turns out that there is a precise equivalence between the extreme and periodic $L_2$-discrepancy which follows from a much broader result about…
We show that the $\mathcal{L}_2$ discrepancy of the explicitly constructed infinite sequences of points $(\boldsymbol{x}_0,\boldsymbol{x}_1, \boldsymbol{x}_2,...)$ in $[0,1)^s$ over $\mathbb{F}_2$ introduced in [J. Dick, Walsh spaces…
It is well known that the two-dimensional Hammersley point set consisting of $N=2^n$ elements (also known as Roth net) does not have optimal order of $L_p$-discrepancy for $p \in (1,\infty)$ in the sense of the lower bounds according to…
The process of symmetrization is often used to construct point sets with low $L_p$ discrepancy. In the current work we apply this method to the shifted Hammersley point set. It is known that for every shift this symmetrized point set…
Low discrepancy point sets have been widely used as a tool to approximate continuous objects by discrete ones in numerical processes, for example in numerical integration. Following a century of research on the topic, it is still unclear…
We start by providing a very simple and elementary new proof of the classical bound due to J. Beck which states that the spherical cap $\mathbb{L}_2$-discrepancy of any $N$ points on the unit sphere $\mathbb S^d$ in $\mathbb{R}^{d+1}$,…