Examples where "thin + thin = nice and thick" 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:


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*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?
 A: 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.
A: I proved this fact not too long ago: if $G$ is a finite group of cardinality $n$, then there exists a subset $S$ of $G$ of cardinality no more than $\lceil 2\sqrt{n\ln n}\rceil$ such that $SS^{-1} = G$. Possibly this is already known ...?
Edit: It IS known, in fact Seva points out in this answer that it has been shown that there exists a subset of size $\lceil \frac{4}{\sqrt{3}} \sqrt{n}\rceil$ satisfying $S^2 = G$. (I still think it's interesting that a probabilistic argument gets us within $\sqrt{\ln n}$ of this. The stronger result relies on the classification of finite simple groups ...)
A: Every real number is the sum of two Liouville numbers, see 
P. Erdős: Representations of real numbers as sums and products of Liouville numbers, Mich. Math. J. 9, 59-60 (1962). ZBL0114.26306. 
So, denoting by $L \subset \mathbb{R}$ the set of Liouville numbers, we have $$\mathbb{R}=L+L.$$
Interestingly, in the the same paper it is also proved that every non-zero real number is the product of two Liouville numbers, so we have an equality for the multiplicative group of the form $$\mathbb{R}^{\times} = L L.$$
Note that $L$ has Lebesgue measure 0, hence it is "thin" with respect to measure theory.
A: Every real number is the sum of two numbers whose continued fraction expansion has no partial quotient exceeding $4$. Marshall Hall, Jr., On the sum and product of continued fractions, Annals of Mathematics, Second Series, Vol. 48, No. 4 (Oct., 1947), pp. 966-993, DOI: 10.2307/1969389, https://www.jstor.org/stable/1969389 
Here, $4$ is best possible. 
A: The set $Q$ of all squares in $\mathbb F_p$ is definitely thick and very nice. Can it be represented as a difference set $A-A$? An open conjecture due to Sárközy is that this is impossible. (It has been recently shown that if $A-A=Q$, then every non-zero element of $Q$ has exactly one representation as $a-b$ with $a,b\in A$, so that $A$ must be thin.) 
