Fix $(L_n)_n$ to be a sequence of lattices, each $L_n\subset \mathbb{R}^n$, where both the effective-inradius and effective-outradius go to 1 (i.e. the Voronoi region of the lattices approach a ball in the appropriate senses, see [1]). Also, for some $r_1,\dots, r_m>1$ take $K_n = \prod_{m=1}^M (r_m\mathcal{B}^n)$ where $\mathcal{B}^n$ is the unit $n$-ball.

Finally, take $(U_n)_n,(x_n)_n$ to be arbitrary sequences of unitary transformations, vectors, respectively, in $\mathbb{R}^{Mn}$. I am interested in approximating the growth of $|(L_n)^M\cap (U_nK_n+x_n)|$ as $n$ grows, i.e. determining the following limit (which I believe exists in general) \begin{equation} \lim_n \frac{1}{n} \log \left(|(L_n)^M\cap (U_nK_n+x_n)|\right) \end{equation}

I believe for such nice lattices and such a simple base set, lattice counting should behave approximately like integration in large dimension, so the limit is eventually like $\lim_n \frac{1}{n}\log(\operatorname{vol}(K_n)/\operatorname{vol}(\mathcal{B}^n)^M) = \log(\prod_mr_m)$ but I cannot substantiate this. Does anyone know any easy paths to this sort of result?