I'm interested in examples where the sum of a set with itself is a substantially bigger set with nice structure. Here are two examples:

**Cantor set**: Let $C$ denote the ternary Cantor set on the interval $[0,1]$. Then $C+C = [0,2]$. There are several nice proofs of this result. Note that the set $C$ has measure zero, so is "thin" compared to the interval $[0,2]$ whose measure is positive.**Goldbach Conjecture**: Let $P$ denote the set of odd primes and $E_6$ the set of even integers greater than or equal to 6. Then the conjecture states is equivalent to $P + P = E_6$. Note that the primes have asymptotic density zero on the integers, so the set $P$ is "thin" relative to the positive integers.

Are there other nice examples?