GivenLet $k,n\in\mathbb{N}$$k,m\in\mathbb{N}$ be given. Let M:={0,... , m-1}$M:=\{0,... , m-1\}$. How to find a subset $T\subset M$, |T|=k with T={ $n_1$,...,$n_k$ | $n_i\in M$ } and T+T|={ (a+b)%m |$|T|=k$ such that $a\in N,b\in N$$|T+T|$ is maximal, where }$T+T=\{ (a+b)\mathbin\%m \mid a\in T,b\in T \}$ ("%"“%” means modulo) such that |T+T| is max.?
I tried to construct a sequence of numbers which maximize |T+T|$|T+T|$. But I couln'tcouldn’t figure out:
- isIs it possible to cover the whole set M$M$ for $k\leq \sqrt{n}$
$k\leq \sqrt{m}$? - What is the best way to construct such a sequence in Theory.theory?
I am looking for papers which deals with this topic or any word to find those papers. I don'tdon’t think this problem is running under the ordinary topicname "settopic name “set covering problems"problems”.
My Ideaidea to construct such a sequence is $a_0=0;a_{i+1}=a_i+(k-i)$ for T={$a_i$, i=0,...,k-1}$T=\{a_i\mid i=0,...,k-1\}$ to get as lesssmall number of collisions as possible collision inamong the sums of |T+T|in $T+T$. But random subsets of M$M$ show me, that there are better subsets.
In my opinion it is hard to find such aan optimal subset T$T$.
Sorry for my bad englishEnglish.
