Let $\Lambda$ be a lattice (discrete additive subgroup) in $\mathbb R^n$ ($n\geq 2$). In my problem, $\Lambda$ lies in a $k$ dimensional ($1< k\leq n$) subspace of $\mathbb R^n$. Let $A\subset \Lambda$ be a subset with cardinality $|A|=m$. To make it concrete, suppose $A$ is the intersection of $\Lambda$ with a hyper cube centered at origin.

My question is how to find a good upper bound on the size of the sumset $A+A$, in terms of $m,k,n$. More generally, on the size of $hA:=A+A+\ldots+A$.

I know there are theories, in particular additive combinatorics, dealing with the size of sumset. However this problem is more specific (we consider the sumset of set with itself and it lies in a subspace, etc.), hence I suspect there may be a tight upper bound.

Since I am not familiar with this filed at all (I am an engineering student), can someone give some hints or pointers? Any comment is welcome!