Let $X\subset \mathbb{R}$ and $Y \subset \mathbb{R}$ where X and Y have finite cardinalities. Let also, $a,b \in \mathbb{R}/0$. What can we say about cardinality $|aX+bY|=????$
For example we can have lower bound and upper bound $\max(|X|,|Y|) \le |aX+bY| \le |X||Y|$. What about more tight bounds?
I am pretty sure one can show that $|aX+bY|=|X||Y|$ for almost all $a,b$, but I don't know how. Thanks, any help will be appreciated.
Also, notation means $aX+bY=\{ax+by: x\in X ,y\in Y \}$.
For fixed $a,b\in\mathbb R$, the map $f_{a,b}\colon X\times Y\to \mathbb R$, $(x,y)\mapsto ax+by$ is injective unless there exist $x_1,x_2\in X$ and $y_1,y_2\in Y$ with $a(x_1-x_2)=b(y_1-y_2)$ and $x_1\ne x_2$ or $y_1\ne y_2$. For each such $x_1,x_2,y_1,y_2$ the set $\{\,(a,b)\in\mathbb R^2\mid a(x_1-x_2)=b(y_1-y_2)\,\}$ describes a line in $\mathbb R^2$. By varying $x_1,x_2,y_1,y_2$ we obtain only finitely many lines, so their union is a set of measure $0$. In this sense, for almost all $(a,b)$ the map $f_{a,b}$ is injective, i.e. $aX+bY$ has precisely $|X|\cdot |Y|$ elements.