Packing rectangles: Does rotation ever help?

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          ^{@YosemiteStan's example.}

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          ^{Detail: Tilt angle $=\sin ^{-1}\left(5 \sqrt{\frac{2}{61}}\right) \approx 65^\circ$.}

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The classic answer to this is a paper of Erdos and Graham 'On packing squares with equal squares'. Given a square of side $n+\varepsilon$, where $0<\varepsilon<1$, we can obviously fit in $n^2$ unit squares, and it's fairly trivial to check that if the squares are axis aligned this is best possible (count the squares intersecting vertical lines on the integers). Obviously, this means about $2\varepsilon n$ area is going to waste.

But Erdos and Graham show one can cover asymptotically all but $n^{7/11}$ area, using skew angles - this is maybe more surprising than Yosemite Stan's example (which also works perfectly well).