Let $k,m\in\mathbb{N}$ be given. Let $M:=\{0,... , m-1\}$. How to find a subset $T\subset M$, $|T|=k$ such that $|T+T|$ is maximal, where $T+T=\{ (a+b)\mathbin\%m \mid a\in T,b\in T \}$ (“%” means modulo)?

I tried to construct a sequence of numbers which maximize $|T+T|$. But I couldn’t figure out:

- Is it possible to cover the whole set $M$ for $k\leq \sqrt{m}$?
- What is the best way to construct such a sequence in theory?

I am looking for papers which deals with this topic or any word to find those papers. I don’t think this problem is running under the ordinary topic name “set covering problems”.

My idea to construct such a sequence is $a_0=0;a_{i+1}=a_i+(k-i)$ for $T=\{a_i\mid i=0,...,k-1\}$ to get as small number of collisions as possible among the sums in $T+T$. But random subsets of $M$ show me that there are better subsets.

In my opinion it is hard to find such an optimal subset $T$.

Sorry for my bad English.