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Anyway, I have done this but am supposed to remark on possible applications. I'm not sure to what I could apply this though. Maybe something to do with ergodics? Any suggestions would be appreciated.If [itex] f:[0,1] \to \mathbb R [/itex] is a twice differentiable function, define [itex] \Gamma = \{y = f(x)\} [/itex] the curve associated to f. Show that the following are equivalent:

- [itex]m(\Gamma+\Gamma)>0[/itex]
- [itex] \Gamma +\Gamma [/itex] contains an open set.
- f is non-linear