A classical theorem of Rogers states that for any convex body $K$ in $n$-dimensional Euclidean space there exists a covering of the space by translates of $K$ with density not exceeding $n\log{n}+n\log\log{n}+5n$. Rogers' theorem does not say anything about the structure of such a covering. We show that for sufficiently large values of $n$ the same bound can be attained by a covering which is the union of $O(\log{n})$ translates of a lattice arrangement of $K$.

MSC Classifications:

52C07, 52C17 show english descriptions Lattices and convex bodies in $n$ dimensions [See also 11H06, 11H31, 11P21]
Packing and covering in $n$ dimensions [See also 05B40, 11H31] 52C07 - Lattices and convex bodies in $n$ dimensions [See also 11H06, 11H31, 11P21]
52C17 - Packing and covering in $n$ dimensions [See also 05B40, 11H31]

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