An upper bound on the minimal dispersion

Abstract

For $\epsilon \in (0,1/2)$ 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 cube $[0,1{]}^d$, which intersects all axis-parallel boxes with volume $\epsilon$. That is, the dispersion of this point set is bounded from above by $\epsilon$.

References in corpus (1)

• A lower bound for the dispersion on the torus

Cited by in corpus (5)

• A note on the dispersion of admissible lattices
• The minimal $k$-dispersion of point sets in high-dimensions
• A tight lower bound on the minimal dispersion
• Lower bounds on the minimal dispersion of point sets via cover-free families
• On the fixed volume discrepancy of the Korobov point sets