Sum of two i.i.d random variables cannot have uniform distribution

Question

My goal is to show that for $U \sim \mathcal{U}[-1,1]$ there are no i.i.d random variables $X,Y$ such that $U = X+Y$. I have seen some arguments why it is impossible, but I wanted to do it with characteristic functions. So the assumptions lead me to the equality $$\varphi_X^2(t) = \frac{\sin{t}}{t}$$ Is there a simple argument why it cannot happen?