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, $j<o(\sqrt{n})$) 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.


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