Looking for references I am looking for reference on the following problem.
Let $S= \{ ax_1+bx_2 \mid x_1 \in  X_1 , x_2 \in X_2, \}$ where $X_1,X_2\subseteq \mathbb{R}^n$ and $a,b \in \mathbb{R}$. Note that $a$ and $b$ are some scalars and $x_1$ and $x_2$ are vectors. 
I am interested in characterizing cardinality of the set $S$ in terms of $a$ and $b$.
For example for the case of $n=1$ when $a,b \in \mathbb{R}/ \mathbb{Q}$ then cardinality of $S$ is $|S|=|X_1||X_2|$. 
Does this statement hold for $n>1$? I would like to know if there exist any literature on this.
Thank you in advance. 

I don't think what you've written down is true for $n=1$. Let $a=\sqrt2$, $b=\sqrt3$, meeting the condition that $a$ and $b$ are irrational. Let $X_1=\{0,\sqrt3\}$, $X_2=\{0,\sqrt2\}$, so the product of the sizes is 4. $S=\{0,\sqrt6,2\sqrt6\}$ has size 3. 
