Show that $X - Y$ is not uniformly distributed on (a,b) for some aLet X and Y be independent and identically distributed. 
Show that $X - Y$ is not uniformly distributed on (a,b) for some  $a<b$.
Let $Z = X - Y$. I know that in order for $Z$ to be uniformly distributed in some interval $(a,b)$ it will be when $a = -b$. And as  X and Y are identically distributed, then they will have the same characteristic function.
I don't really know how to keep going using characteristic functions. I'd appreciate any hint! Thanks.

The characteristic function of uniform distribution on $(-a,a)$ is $\frac  1 {2a}\int_{-a}^{a} e^{itx} dx=\frac {\sin (at)} {at}$. Note that this function takes both positive and negative values. However $Ee^{it(X-Y)}=|Ee^{itX}|^{2} \geq 0$. Hence $X-Y$ cannot be uniformly distributed . (I have used your observation the interval on which we have uniform distribution has to be symmetric).
