An additive combinatorics problem

Question

Given $n,m\in\Bbb N$.

We want to find two disjoint sets $A$ and $B$ such that $$|A|=|B|=n$$ $$\min\{a\in A,b\in B\}>m$$ $$|A+B|=2n$$ where $A+B=\{a+b:a\in A, b\in B\}$.

What is the minimum possible value for $\max\{a\in A,b\in B\}$?

Answer 1

I assume we're interested in nonempty sets, so $n\geq1$. The conditions $|A|=|B|=n$ and $|A+B|=2n$ further require $n\geq2$. I'll need this to define $B$.

The minimum possible value of $\max(A\cup B)$ is $m+n+1$. It is achieved (not uniquely) by $A=[m+1,m+n]$ and $B=[m+1,m+n-1]\cup\{m+n+1\}$, so that $A+B=[2m+2,2m+2n+1]$.

The only other possibility for the first two conditions is $A=B=[m+1,m+n]$, but then as Element118 noted, $|A+B|=2n-1$, which violates the third condition.