TITLE: Size of sets with complete double
QUESTION [19 upvotes]: Let $[n]$ denote the set $\{0,1,...,n\}$. A subset $S\subseteq [n]$ is said to have complete double if $S+S=[2n]$. Let $m(n)$ be the smallest size of a subset of $[n]$ with complete double. My questions are:

Question 1: has this function been studied before? Any reference? It looks related to Sidon sets, but not quite the same.
Question 2: It is easy to see that $m(n)\geq 2\sqrt{n}-1$. Can we show that $m(n)\approx 2\sqrt{n}$ asymptotically? (see the comments on how to get to $2\sqrt{2n}$).

I found one similar question (for $\mathbb F_2^n$ instead of integers) here but could not find further useful references.

REPLY [4 votes]: Short answer: Such a set $S$ is known as a restricted additive basis.

As observed in comments, the general term to search for is [finite]
additive basis: a finite set $S \subset \mathbb{N}$ such that $S+S
\supseteq [2n]$.  Another term is postage stamp problem (not to
be confused with the Frobenius postage stamp problem which is
different).
The extra condition that $S \subseteq [n]$ makes this the
restricted postage stamp problem and the solutions are restricted
additive bases. ("Restricted" means that the elements are restricted
to be in $[n]$.)
It is then (almost) equivalent to ask

given $k$, what is the largest $n$ ...
given $n$, what is the smallest $k$ ...

such that there exists a restricted additive basis of $k$ elements for
the interval $[2n]$.  More about that "almost" at the end of this
answer.  Let us write $n(k)$ for the largest $n$ when $k$ is given,
and $k(n)$ for the smallest $k$ when $n$ is given.
Here the question asks for $k(n)$.  For $n(k)$ this is OEIS
A006638, where currently the largest entry is $a(47)=734$ by
yours truly (2015).  Some translation needed: That "47" counts only the
nonzero elements, so in fact there are 48 elements; that "734"
means the target interval is $[2n]=[734]$, so in our current notation
it becomes $n(48)=367$, and $k(367)=48$.

To the specific questions:
Q1. The OEIS entry gives some pointers to literature.
The oldest reference that I know is H. Rohrbach, "Ein Beitrag zur additiven
Zahlentheorie", Math. Z., 42 (1937), 1-30, which studies both the
unrestricted and the restricted versions.
Symmetric bases have also been studied, see for example S. Mossige,
"Algorithms for computing the h-range of the postage stamp problem",
Math. Comp. 36 (1981), 575–582.
Q2. About asymptotics: No, one cannot achieve $k(n) \sim
2\sqrt{n}$ asymptotically.  Gang Yu, "Upper bounds for finite additive
2-bases", Proc. AMS 137 (2009), 11-18, shows that
$$\limsup_{n \to \infty} \frac{n}{k_r(n)^2} \le 0.419822,$$
where $k_r(n)$ is the smallest size of a restricted basis for $[n]$.
I believe this is currently the best lower bound for $k_r$.
Translated to our notation (with target interval $[2n]$)
this means the factor in front of $\sqrt n$ is asymptotically at least $2.18264$.

And about the "almost": Generally $k(n)$ increases as $n$
increases, but there are some known cases where $k(n)$ suddenly steps
down. One of them was also noticed in the comments above: "What the
heck is going on at n=31?".  Indeed we have $k(31)=14$ but $k(32)=13$;
$k(51)=18$ but $k(52)=17$; and $k(57)=19$ but $k(58)=18$.
See Table 5 in this 2018 paper by me and coauthors, where the
corresponding problem is studied in two dimensions.