We demonstrate that every difference set in a finite Abelian group is equivalent to a certain ‘regular’ covering of the lattice $ A_n = \{ \boldsymbol {x} \in \mathbb {Z} ^{n+1} : \sum _{i} x_i = 0 \} $ with balls of radius $ 2 $ under the $ \ell _1 $ metric (or, equivalently, a covering of the integer lattice $ \mathbb {Z} ^n $ with balls of radius $ 1 $ under a slightly different metric). For planar difference sets, the covering is also a packing, and therefore a tiling, of $ A_n $. This observation leads to a geometric reformulation of the prime power conjecture and of other statements involving Abelian difference sets.

Let $p$ be a prime, and let ${{\zeta }_{p}}$ be a primitive $p$ -th root of unity. The lattices in Craig's family are $(p\,-\,1)$ -dimensional and are geometrical representations of the integral $\mathbb{Z}[{{\zeta }_{p}}]$ -ideals ${{\left\langle 1\,-\,{{\zeta }_{p}} \right\rangle }^{i}}$ , where $i$ is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions $p\,-\,1$ where $149\,\le \,p\,\le \,3001$ , Craig's lattices are the densest packings known. Motivated by this, we construct $(p\,-\,1)(q\,-\,1)$ -dimensional lattices from the integral $\mathbb{Z}[{{\zeta }_{pq}}]$ -ideals ${{\left\langle 1\,-\,{{\zeta }_{p}} \right\rangle }^{i}}{{\left\langle 1\,-\,{{\zeta }_{q}} \right\rangle }^{j}}$ , where $p$ and $q$ are distinct primes and $i$ and $j$ are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.