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?

[1]: https://ieeexplore.ieee.org/document/1512416

  • $\begingroup$ The problem arises in calculation of an entropy for a network information theory problem. $\endgroup$ – enthdegree Dec 29 '18 at 0:30
  • $\begingroup$ Ok one way you might be able to do it it is if you replace the fixed lattice sequence $(L_n)_n$ with a sequence of random lattices, each one taken from a Minkowski-Hlawa-Siegel ensemble that also has the aforementioned properties. But this is horrifically ugly. $\endgroup$ – enthdegree Dec 29 '18 at 0:55
  • $\begingroup$ It holds on average over all lattices of a fixed determinant, due to results by Rogers. The proof is a bit involved, I will publish it and replicate it here after peer review! $\endgroup$ – enthdegree Jan 7 at 4:01

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