Possible Applications of Proving Non-Linear Functions on Curves

[Kreizhn]

I was given a problem by a professor to prove the following problem:

If $f:[0,1] \to \mathbb R$ is a twice differentiable function, define $\Gamma = \{y = f(x)\}$ the curve associated to f. Show that the following are equivalent:

1. $m(\Gamma+\Gamma)>0$
2. $\Gamma +\Gamma$ contains an open set.
3. f is non-linear

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.

 

[Kreizhn]

Any suggestions?

 

[HallsofIvy]

I take it you mean that $\Gamma$ is the set $\{(x, y)| y= f(x)\}$.

What do you mean by $\Gamma+ \Gamma$? How are you adding sets?

 

[Kreizhn]

Yes, that is what I meant. Sorry for the sloppiness, though I believe it's not an uncommon shorthand.

Set addition is taken to be naive: nothing special like essential sums. So
$$A+B = \{ a+b: a \in A, b \in B\}$$

 

[Kreizhn]

Anybody have any ideas?