Related papers: The inverse of the star-discrepancy problem and th…
The star-discrepancy is a quantitative measure for the irregularity of distribution of a point set in the unit cube that is intimately linked to the integration error of quasi-Monte Carlo algorithms. These popular integration rules are…
By a profound result of Heinrich, Novak, Wasilkowski, and Wo{\'z}niakowski the inverse of the star-discrepancy $n^*(s,\ve)$ satisfies the upper bound $n^*(s,\ve) \leq c_{\mathrm{abs}} s \ve^{-2}$. This is equivalent to the fact that for any…
By a result of Heinrich, Novak, Wasilkowski and Wo\'zniakowski the inverse of the star discrepancy $n(d,\varepsilon)$ satisfies $n(d,\varepsilon)\leq c_{\abs}d\varepsilon^{-2}$. Equivalently for any $N$ and $d$ there exists a set of $N$…
We show that there is a constant $K > 0$ such that for all $N, s \in \N$, $s \le N$, the point set consisting of $N$ points chosen uniformly at random in the $s$-dimensional unit cube $[0,1]^s$ with probability at least $1-\exp(-\Theta(s))$…
In 2001 Heinrich, Novak, Wasilkowski and Wo\'zniakowski proved that the inverse of the star discrepancy satisfies $n(d,\varepsilon)\leq c_{\abs}d \varepsilon^{-2}$ by showing that there exists a set of points in $[0,1)^d$ whose…
The inverse of the star discrepancy, $N(\epsilon, s)$, defined as the minimum number of points required to achieve a star discrepancy of at most $\epsilon$ in dimension $s$, is known to depend linearly on $s$. However, explicit…
The star discrepancy is a quantitative measure of the uniformity of a point set in the unit cube. A central quantity of interest is the inverse of the star discrepancy, $N(\varepsilon, s)$, defined as the minimum number of points required…
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…
The star discrepancy $D_N^*(\mathcal{P})$ is a quantitative measure for the irregularity of distribution of a finite point set $\mathcal{P}$ in the multi-dimensional unit cube which is intimately related to the integration error of…
The $L_{\infty}$ star discrepancy is a very well-studied measure used to quantify the uniformity of a point set distribution. Constructing optimal point sets for this measure is seen as a very hard problem in the discrepancy community.…
In 2004 the second author of the present paper proved that a point set in $[0,1]^d$ which has star-discrepancy at most $\varepsilon$ must necessarily consist of at least $c_{abs} d \varepsilon^{-1}$ points. Equivalently, every set of $n$…
The inverse of the star-discrepancy $N^*(d,\ve)$ denotes the smallest possible cardinality of a set of points in $[0,1]^d$ achieving a star-discrepancy of at most $\ve$. By a result of Heinrich, Novak, Wasilkowski and Wo{\'z}niakowski, $$…
Low discrepancy point sets have been widely used as a tool to approximate continuous objects by discrete ones in numerical processes, for example in numerical integration. Following a century of research on the topic, it is still unclear…
The $L_{\infty}$ star discrepancy is a measure for the regularity of a finite set of points taken from $[0,1)^d$. Low discrepancy point sets are highly relevant for Quasi-Monte Carlo methods in numerical integration and several other…
In the present paper we prove several results concerning the existence of low-discrepancy point sets with respect to an arbitrary non-uniform measure $\mu$ on the $d$-dimensional unit cube. We improve a theorem of Beck, by showing that for…
According to Aistleitner and Weimar, there exist two-dimensional (double) infinite matrices whose star-discrepancy $D_N^{*s}$ of the first $N$ rows and $s$ columns, interpreted as $N$ points in $[0,1]^s$, satisfies an inequality of the form…
For $m, d \in {\mathbb N}$, a jittered sampling point set $P$ having $N = m^d$ points in $[0,1)^d$ is constructed by partitioning the unit cube $[0,1)^d$ into $m^d$ axis-aligned cubes of equal size and then placing one point independently…
Geometric discrepancies are standard measures to quantify the irregularity of distributions. They are an important notion in numerical integration. One of the most important discrepancy notions is the so-called \emph{star discrepancy}.…
We study the extreme and the periodic $L_p$ discrepancy of point sets in the $d$-dimensional unit cube. The extreme discrepancy uses arbitrary sub-intervals of the unit cube as test sets, whereas the periodic discrepancy is based on…
For all $s \geq 1$ and $N \geq 1$ there exist sequences $(z_1,\ldots,z_N)$ in $[0,1]^s$ such that the star-discrepancy of these points can be bounded by $$D_N^*(z_1,\ldots,z_N) \leq c \frac{\sqrt{s}}{\sqrt{N}}.$$ The best known value for…