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Rotated squares

John pointed me out that there is conjecture of Toeplitz from 1911, that every simple closed curve on the plane contains an inscribed square. It is open but it was proved by Stromquist in 1989 that if the curve is locally the graph of a continuous function near any of its points, then the conjecture is true.

Could this be true if the closed curve is replaced by closed set Hausdorff dimension slightly less then 2? Or if the set has positive upper density on the plane (or even a higher dimensional Euclidean space) does it contain a square of every size bigger than a fixed number?

 

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