Let $V:= \mathbb R^n$ with the standard scalar product. Further, let $L \subset V$ be a lattice of rank $n$ and let $W \subset V$ be a subspace of dimension $m<n$ such that $\pi(L)$ is dense in $W$ where $\pi:V \to W$ is the orthogonal projection onto $W$ (so $\ker(\pi) = W^\perp$ and $\pi(V)=W$). For safety let's also assume that $\pi$ is injective when you restrict it to the lattice $L$.
Now let $A \subset W$ be open and bounded and let $(a_n)_{n \in \mathbb N}$ a sequence of lattice points of $L$ running through all points which project to $A$. So we have $$L \cap \pi^{-1}(A) = \{a_n : n \in \mathbb N\}.$$ Further, we sort them such that the sequence $||a_n-\pi(a_n)||$ is monotone increasing (the geometric meaning of $||a_n-\pi(a_n)||$ is just the distance from $a_n$ to $W$).
I have two conjetures.
Conjecture 1: The sequence $\pi(a_n)$ behaves like picking points from a continuous uniform distribution on $A$. With that I mean that for every open $B \subset A$ we have $$\lim_{n \to \infty} \frac{\#\{j \le n : \pi(a_j) \in B\}}{n} = \frac{\lambda(B)}{\lambda(A)}.$$ Here $\lambda$ is the Lebesgue messure on $W$.
Conjecture 2: The number of points in $\pi^{-1}(A)$ grows accordingly too the volume of the lattice. With that I mean $$\lim_{n \to \infty} \frac{V(||a_n-\pi(a_n)||) \cdot \lambda(A)}{n \cdot \operatorname{vol}(L)}=1. $$ Here with $V(r)$ we denote the volume of an $m$-dimensional sphere with radius $r$.
I guess my two conjectures are very natural and say just what you would expect. So if they are true I'm quite certain that proofs already exist in the literature. Do you have references or can you proof them yourself? I'm also happy with both.