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. 
 A: You mentioned cardinality, so I will assume $X_1 , X_2$ are finite. I will start with the easier case where $a, b \in \mathbb{Q}$ and then move onto the case where $a$ and $b$ are algebraic numbers. I will discuss finding lower bounds for $|S|$ in terms of $|X_1|$ and $|X_2|$, as in general $|S|$ can be as large as $|X_1||X_2|$. Consider the case $n = 1$.
When $a$ and $b$ are in $\mathbb{Q}$, one can clear denominators of $a$ and $b$ without changing your proposed problem so that $a , b \in \mathbb{Z}$. After this, divide by $\gcd(a,b)$ so we may assume $a$ and $b$ are coprime. Let $[1,N] = \{1 , \ldots , N\}$. In this case taking $X_1 = b \cdot [1,N]$ and $X_2 = a \cdot [1,M]$, it is easy to check that $|S| = |X_1| + |X_2| - 1$. This shows that $|S|$ can be as small as possible. If one strengthens your hypothesis so that $X_1 = X_2$, it has been recently shown using elementary methods that $|S| \geq (|a|+|b|)|X_1| - C_{a,b}$ where $C_{a,b}$ is a constant only depending on $a$ and $b$. This is the best possible up to the additive constant by observing what happens when $X_1 = [1,N ]$. The reference is here http://arxiv.org/pdf/1311.0422.pdf and see the references of this for past work on the problem. In particuar Boris Bukh showed the analogous result for higher order dilates with a weaker error term in his Sum of Dilates paper http://arxiv.org/abs/0711.1610. 
Now it seems as you may be more interested in the case where $a$ and $b$ are algebraic numbers, but indeed it is instructive to first understand the easier case where $a,b \in \mathbb{Q}$. Suppose $X_1 , X_2 \subset \mathbb{Q}(a,b)$ and let $d = [\mathbb{Q}(a,b) : \mathbb{Q}]$. Think of $a$ and $b$ as linear transformations over $\mathbb{Q}$ of degree $d$ in the natural way and similarly think of $X_1, X_2$ as a subset of $\mathbb{Q}^d$ in the natural way. By clearing denominators, we may assume $X_1 , X_2 \subset \mathbb{Z}^d$. Problem 5 here: http://www.borisbukh.org/problems.html gives a conjecture for the lower bound of $|S|$ when $X_1 = X_2$. When $X_1 \neq X_2$, the best result I know in this direction is an exercise in Tao in Vu's book on additive combinatorics: exercise 3.4.7 which the proof can be found in Lemma 2.4 in http://arxiv.org/pdf/math/0511069v2.pdf. 
When considering $a$ and $b$ algebraic integers one is naturally lead to Brunn-Minkowski inequality from real analysis. I will roughly indicate the connection, which is indicated by the exercise in Tao and Vu I mention above. As above, we may think of $X_1$ and $X_2$ as subsets of $\mathbb{Z}^d$. Given convex sets $A,B \subset \mathbb{R}^d$ and a lattice $\Gamma$, consider $A \cap \Gamma$ and $B \cap \Gamma$. Place a unit box around each lattice point in $A \cap \Gamma$ and call the union of these unit boxes $U$ and places a unit box around each lattice point of $B \cap \Gamma$ and call the union of these unit boxes $V$. Then one can check the Lebesgue measure of $U + V$ (Minkowski sum) closely approximates $|(A \cap \Gamma) + (B \cap \Gamma)|$. The Lesbegue measure of $U+V$ can be estimated using Brunn-Minkowski. One may expect that to minimize $|X_1 + X_2|$, one should choose $X_1$ and $X_2$ to be the lattice points in some convex sets. 
A: 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. 
