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Restricted progressions

We have finished with Brian an updated version of our paper on restricted arithmetic progressions over finite fields. It is available at: http://arxiv.org/abs/1011.5302 .

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Behrend's example

It turns out that Behrend's construction also provides examples of sets A\subseteq [1,N] of size |A|\geq N\,e^{-c_k\sqrt{log\,N}}, which does not contain any non-trivial solution of the equation

x_1+x_2+...+x_k=k\, x_{k+1}

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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.

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