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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

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

This is because if k points are on a sphere then their arithmetic mean cannot be on the same sphere unless they are all equal. Are there essentially better upper bounds for the size of such sets A for large k, then for k=2 that is for 3-term arithmetic progressions?

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