A Note on Rectangle Covering with Congruent Disks

In this note we prove that, if $S_n$ is the greatest area of a rectangle which can be covered with $n$ unit disks, then $2\leq S_n/n<3 \sqrt{3}/2$, and these are the best constants; moreover, for $Δ(n):=(3\sqrt{3}/2)n-S_n$, we have $0.727384<\liminfΔ(n)/\sqrt{n}<2.121321$ and $0.727384<\limsupΔ(n)/\sqrt{n}<4.165064$.

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