Given an integer lattice, how to count the number of points whose norm is smaller than some bound $M$? Let $\mathbf{b}_1, \mathbf{b}_2, ..., \mathbf{b}_n$ be linearly independent  $m$-dimensional vectors whose entries belong to $[0, M] \cap \mathbb{Z}$, for some $M \in \mathbb{N}^*$. Of course, $n \le m$ and I am mainly interested in the case $n < m$.
Let $L$ be the integer lattice whose basis is formed by these vectors. Hence, any $\mathbf{u} \in L$ can be written as $\sum_{i=1}^n c_i\mathbf{b}_i$ for $c_i \in \mathbb{Z}$.
Is there some method to derive upper-bounds to the number of points of $L$ that have norm smaller than or equal to $M$?
That is, an upper-bound to the cardinality of the following set:
$$\left\{ \mathbf{u} \in L : ||\mathbf{u}|| \le M \right\}$$
Here, $||\mathbf{u}|| := \sqrt{\sum_{i=1}^n u_i^2}$, that is, the Euclidean norm.
 A: Let $b_1,\dots,b_n$ in $[-M,M]^m$ be linearly independent, and let $L = \sum_i \mathbb{Z}b_i$ be the lattice they generate.
For $i = 1, \dots,n$, let $r_i$ be the smallest positive real number such that $L$ contains $i$ linearly independent vectors of $\ell^\infty$-norm $\leq r_i$. We have
$$
0 < r_1 \leq r_2 \leq \dots \leq r_n \leq M .
$$
By a result of Henk (th 2.1 in this note by Boucksom), we have for any $c \geq 1$ the inequality
$$
| \{ b \in L \ | \ ||b||_{\infty} \leq cM \ \}| \leq 2^n \prod_{i=1}^n \left(  \frac{2cM}{r_i} + 1\right) \leq (2(2c+1)M)^n \prod_{i=1}^n r_i^{-1}.
$$
On the other hand, a theorem of Minkowski (th 4.1 in Boucksom's note) yields
$$
 \prod_{i=1}^n r_i^{-1} \leq \frac{n!}{2^n} V,
$$
where $V$ is the euclidean volume of 
$$
\{ (x_1,\dots,x_n) \in \mathbb{R}^n \ | \ ||\sum_{i=1}^n x_i b_i||_{\ell^\infty} \leq 1 \}.
$$
Thus we get
$$
| \{ b \in L \ | \ ||b||_{\infty} \leq cM \ \}| \leq n!(2(2c+1)M)^n V,
$$
for $c \geq 1$, and in particular an upper bound for your question is given by $n! (6M)^n V$. If the $b_i$'s are small, then $V$ is large (and so is the LHS).
A: (I'm operating under the assumption that you want an upper bound that holds for all such lattices. If you have a specific lattice that interests you, then there are potentially other methods to try.)
For an easy bound, you can just use basic packing arguments to see that no rank $n$ lattice can have more than, say, $(2M+1)^n$ points of length at most $M \cdot \lambda_1(L)$, where $\lambda_1(L)$ is the length of the shortest lattice vector, which for you is at least $1$. In particular, assume without loss of generality that the lattice lies in $\mathbb{R}^n$ (by viewing the lattice as embedded in its span), and place an $n$-dimensional ball of radius $\lambda_1(L)/2$ around each lattice vector. Notice that these balls are disjoint and that they all lie in a ball of radius $M+1/2$. Therefore, the total volume of the small balls must be at most the volume of the larger ball, which immediately yields the bound.
For stronger bounds with a much more difficult proof, first notice that the densest such lattice "should" simply be $\mathbb{Z}^n$. This is morally true because $\mathbb{Z}^n$ has the minimal determinant amongst all such lattices. Since a "typical" set of volume $V$ in the subspace spanned by your lattice has $V/\det(L)$ points, the lattice with smallest determinant tends to have more points inside of it. E.g., this provably holds as $M \to \infty$.
I'm not sure if $\mathbb{Z}^n$ is exactly the densest such lattice for all $n,m, M$ (some related conjectures are false). But, we know that something close to this is true. In particular, in work with Regev, we showed that no lattice $L \subseteq \mathbb{R}^n$ whose sublattices $L' \subseteq L$ all satisfy $\det(L') \geq 1$ has "significantly'' more points than $\mathbb{Z}^n$. Your lattices clearly satisfy this constraint, so the bound applies. Here's the precise statement, and there are of course more details in the paper:

I hope that the notation is clear. We write $rB_2^n$ for an $n$-dimensional ball of radius $n$, and you should think of our $\mathcal{L}$ as your lattice embedded in the subspace that it spans.
(You can almost certainly get tighter bounds for the specific class of rank $n$ lattices that are sublattices of $\mathbb{Z}^m$ (the restriction on the coordinates of the basis seems unlikely to matter much), but this might be sufficient for you.)
