TITLE: Examples where "thin + thin = nice and thick"
QUESTION [57 upvotes]: 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?

REPLY [7 votes]: I know you asked for examples of the "thin + thin = nice and thick" phenomenon but, since Palindrome Week is all the rage these days, I can't avoid mentioning the following example of "thin + thin + thin = nice and thick".
A couple of years ago, J. Cilleruelo(†), F. Luca, and L. Baxter proved that every natural number $n$ can be written as a sum of three palindromic numbers. Since the natural density of the set of palindromic numbers is $0$, if we agree to regard $\mathbb{N}$ as "nice" and "thick", we do have here--as promised--an example of the "thin + thin + thin = nice and thick" phenomenon.