Suppose, we have to independent uniform random variables $V \in [0,a]$ and $W\in [0,b]$.
Are there any upper bounds on \begin{align} P [ |U-V| \le r ] \end{align} for some given $r$?
Note that lower bounds are easy since \begin{align} P[|U-V|\le r]=1- P[|U-V|> r] \end{align}
where by Azuma-Hoeffing inequality we can \begin{align} P[|U-V|> r] \le 2 e^{-\frac{r^2}{2 d^2 }} \end{align} where $d$ is given by $ \max |U-V| \le d=\max(a,b)$.