math.MG — Metric Geometry

Multi-coverings by lattice translates of convex bodies

Contributed by Scott Kominers

We show that for every convex body $K\subset\mathbb R^n$ ($n\ge2$), every $0<\varepsilon<1$, and every prescribed density $\rho\ge A_\varepsilon n\log n$, there is an arrangement of translates of $K$ along a single lattice with mean multiplicity exactly $\rho$ and multiplicity at least $(1-\varepsilon)\rho$ everywhere. Consequently, lattice $k$-fold coverings exist with density at most $\max\{(1+\varepsilon)k,C_\varepsilon n\log n\}$, where $C_\varepsilon$ depends only on $\varepsilon$. In sufficiently large dimension $n$, a version with prescribed common covolume holds simultaneously for up to $\exp(n^{c_0})$ bodies whose uniform distributions have a common covariance matrix, with $c_0>0$ absolute. The proof builds on OpenAI's horizontal–vertical construction for single-lattice coverings and uses Gaussian layer averaging and Laplace bounds for covering counts.