相关论文: A note on the dispersion of admissible lattices

The dispersion of a point set in the unit square is defined to be the area of the largest empty axis-parallel box. In this paper we are interested in the dispersion of lattices in the plane, that is, the supremum of the area of the empty…

In this note we present a construction which improves the best known bound on the minimal dispersion of large volume boxes in the unit cube. Let $d>1$. The dispersion of $T \subset [0,1]^d$ is defined as the supremum of the volume taken…

For $\varepsilon\in(0,1/2)$ and a natural number $d\ge 2$, let $N$ be a natural number with \[ N \,\ge\, 2^9\,\log_2(d)\, \left(\frac{\log_2(1/\varepsilon)}{\varepsilon}\right)^2. \] We prove that there is a set of $N$ points in the unit…

We provide a new construction for a set of boxes approximating axis-parallel boxes of fixed volume in $[0, 1]^d$. This improves upper bounds for the minimal dispersion of a point set in the unit cube and its inverse in both the periodic and…

The dispersion of a point set in $[0,1]^d$ is the volume of the largest axis parallel box inside the unit cube that does not intersect with the point set. We study the expected dispersion with respect to a random set of $n$ points…

We study the dispersion of point sets in the unit square; i.e. the size of the largest axes-parallel box amidst such point sets. It is known that $\liminf_{N\to\infty} N\mathrm{disp}(N,2)\in \left[\frac54,2\right],$ where…

We consider the volume of the largest axis-parallel box in the $d$-dimensional torus that contains no point of a given point set $\mathcal{P}_n$ with $n$ elements. We prove that, for all natural numbers $d, n$ and every point set…

We show that, for every set of $n$ points in the $d$-dimensional unit cube, there is an empty axis-parallel box of volume at least $\Omega(d/n)$ as $n\to\infty$ and $d$ is fixed. In the opposite direction, we give a construction without an…

Using, and extending, striking inequalities by V.V. Ivanov on the down-crossings of monotone functions and ergodic sums, we give universal bounds on the probability of finding oscillations of observables in 1-dimensional lattice gases in…

We study the dispersion of a point set, a notion closely related to the discrepancy. Given a real $r\in (0,1)$ and an integer $d\geq 2$, let $N(r,d)$ denote the minimum number of points inside the $d$-dimensional unit cube $[0,1]^d$ such…

We elaborate on the intimate connection between the largest volume of an empty axis-parallel box in a set of $n$ points from $[0,1]^d$ and cover-free families from the extremal set theory. This connection was discovered in a recent paper of…

For any natural number $d$ and positive number $\varepsilon$, we present a point set in the $d$-dimensional unit cube $[0,1]^d$ that intersects every axis-aligned box of volume greater than $\varepsilon$. These point sets are very easy to…

For a positive integer $d$, the $d$-dimensional Chebyshev-Frolov lattice is the $\mathbb{Z}$-lattice in $\mathbb{R}^d$ generated by the Vandermonde matrix associated to the roots of the $d$-dimensional Chebyshev polynomial. It is important…

We give the first nontrivial upper and lower bounds on the maximum volume of an empty axis-parallel box inside an axis-parallel unit hypercube in $\RR^d$ containing $n$ points. For a fixed $d$, we show that the maximum volume is of the…

We study the error of the number of points of the lattice $\mathbb{Z}^{d}$ that fall into a dilated and translated hypercube centred around $0$ and whose axis are parallel to the axis of coordinates. We show that if $t$, the factor of…

In the course of classifying generic sparse polynomial systems which are solvable in radicals, Esterov recently showed that the volume of the Minkowski sum $P_1+\dots+P_d$ of $d$-dimensional lattice polytopes is bounded from above by a…

We study the size of the largest rectangle containing no point of a given point set in the two-dimensional torus, the dispersion of the point set. A known lower bound for the dispersion of any point set of cardinality $n\ge 2$ in this…

The dispersion of a point set $P\subset[0,1]^d$ is the volume of the largest box with sides parallel to the coordinate axes, which does not intersect $P$. Here, we show a construction of low-dispersion point sets, which can be deduced from…

The problem of finding the largest empty axis-parallel box amidst a point configuration is a classical problem in computational geometry. It is known that the volume of the largest empty box is of asymptotic order $1/n$ for $n\to\infty$ and…

For a point set of $n$ elements in the $d$-dimensional unit cube and a class of test sets we are interested in the largest volume of a test set which does not contain any point. For all natural numbers $n$, $d$ and under the assumption of a…