# The typical amount of lattice points in a set as dimension goes off to infinity

I have encountered a geometry-of-numbers-type problem when calculating an entropy in a lattice communications scheme:

$$L^{(n)}$$ is a lattice uniform over all those in $$\mathbb{R}^n$$ with base volume 1, and $$A^{(n)}:=U(x+\prod_{k=1}^K\frac{r_k}{\sqrt[n]{V(B^{(n)})}} B^{(n)})$$ where $$x\in \mathbb{R}^{Kn}, r_k> 1,\ U$$ is some unitary transform over $$\mathbb{R}^{Kn}$$ and $$B^{(n)}$$ is the unit n-ball. Does $$\frac{1}{n}\log|(L^{(n)})^K\cap A^{(n)}|$$ converge in probability towards some value ?

I think it converges to $$\log|\prod_{k=1}^K r_k|$$, but can't prove it.

I believe results by CA Rogers on counting lattice points in sets can be used to give bounds on the first $$O(\sqrt[3]{n})$$ moments of $$|(L^{(n)})^K\cap A^{(n)}|$$. The $$j$$-th moment is lower-bounded by the volume of $$A^{(n)}$$ raised to the $$j$$-th power. Upper bounds are more difficult. My strategy so far has been to try and first cover $$A^{(n)}$$ with translates of hypersphere products $$s=(B^{(n)})^K$$ and then upper bound the moments of the sum of lattice points. For the $$j$$-th moment (again, $$j) my estimation only yields an upper bound of $$V(A^{(n)})O(n\log n)$$. These bounds are not precise enough to establish the limit as far as I know, either because the bounds themselves are not strong or because we only get the very first few moments.