New bounds on the density of lattice coverings

We obtain new upper bounds on the minimal density $\Theta _{n, \mathcal {K}}$ of lattice coverings ${\mathbb {R}}^n$ by dilates a convex body $\mathcal {K}$. also probability (with respect to natural Haar-Siegel measure space lattices) that randomly chosen $L$ satisfies $L+\mathcal {K}= {\mathbb {R}}^n$. As step in proof, we utilize and strengthen results discrete Kakeya problem.