There's a famous problem in probability: find a probability distribution such that two such independent random variables have a sum that is uniformly distributed from 0 to 1.
That turns out to be impossible. One way to prove it is by considering the expected value of exp(2*pi*i*x). Because the distribution of the sum has to have this quantity be 0, each part also has to have the same property. The only distribution with support between 0 and .5 with this property is 0 w.p .5, and 0.5 w.p. 0.5. That distribution doesn't work.
A related problem is: find a probability distribution such that two such independent random variables x1, and x2 satisfy x1 + 2*x2 is uniformly distributed from 0 to 1.
Can you do it?