Cardinality of sum-set of two arithmetic progressions

Question

Suppose that sum-set $A+B$ between sets $A$ and $B$ is defined as $A+B=\{a+b|a \in A, b \in B \}$. We further assume that $A=\{d_az|z \in \mathbb{Z} \}$ and $B=\{d_bz|z \in \mathbb{Z} \}$ where $d_a$ and $d_b$ are some constants in $\mathbb{R}$. My question is under what conditions
$|A+B|=\max(|A|,|B|)$ where $| \cdot |$ denotes cardinality?

Answer 1

If $A,B$ are both countable then $A+B$ is also countable because ${\mathbb N} \times {\mathbb N}$ is countable. If either $A$ and $B$ is infinite then it is easy to show that $|A \times B| = |\max(|A|,|B|)|$ so again you get $|A + B| = \max(|A|,|B|)$. If $A,B$ are both finite then it is more tricky. It should be that $|A + B| = \max(|A|,|B|)$ if and only if either $|A| = 1$ or $|B| = 1$. To see this, assume $|A| \geq |B| > 1$ and take the minimum element of $B$ added with the minimum element of $A$, and then take the maximum element of $B$ added with every element of $A$. You'll get at least $|A| + 1$ distinct elements in $|A + B|$, so $|A + B| > \max(|A|,|B|)$