Combinatorics problem on the size of A+B

Question

Let $A$, $B$ be finite subsets of $\mathbb{Z}$ with $|A|=n$, $|B|=m$. Denote $A+B=\{a+b:a \in A, b \in B\}$. It's fairly easy to show that $|A+B| \geq n+m-1$. My question is:

If $|A+B|=n+m-1$, does it follow that $A \cap B \neq \emptyset$?

Also, could someone give me a reference to a book that deals with this kind of problems (i.e., that gives interesting results on the size of finite sets like $A+B$, $AB$, etc.)?

Thank you!

Answer 1

Let $A_k=\{a+k\mid a\in A\}$. Show that $|A_k+B|=|A+B|$. So your statement is false by picking sufficiently large $k$ so that $k+ \min A> \max B$.

So if you have a pair $|A|=n,|B|=m$ with $|A+B|=n+m-1$ then there is such a pair that is disjoint.

Answer 2

As for references: The books

• Additive Number Theory: Inverse Problems and the Geometry of Sumsets (Nathanson)

• Additve Combinatorics (Tao and Vu)

are standard introductions.

Nathanson's book characterizes sets of integers $A$, $B$, having $|A+B| = |A| + |B|-1$.