Sum of two random variables uniformly distributed on circles Suppose we have two independent random variables $U_1$ and $U_2$ unfiorm  on 
\begin{align}
S_i = \left\{ (s_1,s_2) \in \mathbb{R}: \sqrt{s_1^2+s_2^2} =r_i \right\} 
\end{align}
respectily. Assume $r_1 \ge r_2$.
Question: How to find the pdf of $U_1+U_2$? 
We know that it would be distributed on 
\begin{align}
S_3 = \left\{ (s_1,s_2) \in \mathbb{R}: r_1-r_2 \le  \sqrt{s_1^2+s_2^2} \le r_1+r_2 \right\} 
\end{align}
In other words, show that the sum of two random variables on the circles results in a random variable distributed on an annulus.  
The question now is how to find the pdf of $U_1+U_2$? 
Can this, for example, be done by using characteristic functions? 
 A: It is not the case.
One way to go about deriving the distribution of the sum is to note that the points may be written $U_i=(r_i\sin\theta_i,r_i\cos\theta_i)$ where $r_i$ are fixed and $\theta_i$ are uniform on $(0,2\pi)$.
Now, write $U=U_1+U_2=(R\sin\theta,R\cos\theta)$. As Kavi has already pointed out, the angle $\theta$ will be uniformly distributed due to rotational symmetry, so we can ignore it and focus on the distribution of $R$.
For $U$ to be uniform on the annulus, $R^2$ would have had to be uniformly distributed between $(r_1-r_2)^2$ and $(r_1+r_2)^2$: just write down the area in the annulus with $R<r$ and divide by the total area expressed in terms of $r^2$.
We now derive the distribution of $R^2$ by use of the relation $R^2=r_1^2+r_2^2+2r_1r_2\cos\phi$ where $\phi=\theta_1-\theta_2$. Note that $\phi$ will also be uniform on $(0,2\pi)$ provided we consider angles modulo $2\pi$.
When $\phi$ is uniform on $(0,2\pi)$, the random variable $X=\cos\phi$ will have probability density $\frac{dx}{\pi\sqrt{1-x^2}}$. So, as you can see, the density is higher around the outer and inner edges of the annulus.
The simplest counter-example comes when $r_1=r_2=r$ and you compute $\Pr[R<r]$. This is the  same as the probability that $\cos\phi<-1/2$, which is $1/3$. If $U$ had been uniform on the disc, it should have been $1/4$.
