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In this note we present a construction which improves the best known bound onthe minimal dispersion of large volume boxes in the unit cube. Let $d>1$. Thedispersion of $T \subset [0,1]^d$ is defined as the supremum of the volumetaken over all axis parallel boxes in the cube which do not intersect $T$. Theminimal dispersion of $n$ points in the cube is defined as the infimum of thedispersion taken over all $T$ such that $|T| = n$. Define the "large volume"regime as the class of all volumes $\frac{1}{4} < r \leq \frac{1}{2}$. Theinverse of the minimal dispersion is denoted as $N(r,d)$. When the volume islarge, the best known upper bound on $N(r,d)$ is of the order $ {(r -\frac{1}{4})}^{-1}.$ The construction presented in this note yields an upperbound given by $$N(r,d) \leq \left \lfloor \frac{\pi}{\sqrt{r - \frac{1}{4}}}\right \rfloor - 3 .$$ Some of our intermediate estimates are sharp given thecondition that $d \geq C_{r}$, where $C_r$ is a positive constant which dependsonly on the volume $r$.

Minimal dispersion of large volume boxes in the cube