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The polynomial form of Roth's theorem (previously mentioned by Quid) implies, for instance, that any set of positive density contains a triple $x$, $x+h$, $x+h^2$ (for $h\neq 0$). Notice that this is equivalent to asking for solutions to the nonlinear equation $b^2+b+a^2=c+2ab$. So results of this form do hold for some nonlinear equations.