real analysis – Does the modulo on a high-dimensional network generally maintain the direction?

We know that for sufficiently large dimensions $ n $ there is a trellis $ L_n $ in $ mathbb {R} ^ n $ whose base region of the Voronoi partition encompasses almost a negligible proportion of a $ (1- varepsilon) $-ball, and is also almost contained in a $ (1+ varepsilon) $-ball, almost a negligible proportion.[1] How does this base region behave like a ball in another way?


We can define "round" and "mod" operations on the space up to a network $ L $ and his score of Voronoi as: $ R_L (v) = arg min _ { ell in L} | v- ell | $ and $ mod_L (v) = v-R_L (v). $

We can also define a "mod ball" as $ mod ^ ast_n (v) = (v / | v |) cdot [|v| – operatorname{round}(|v|/2)]$, which reduces $ v $ in lengths of 2 until it is in the ball.

Fixing radius $ R> $ 0, for a fairly high dimension $ n $, can there be a trellis $ L $ in this dimension with $ | mod_ ast (v) – operatorname {mod} _L (v) | < varepsilon $ for the majority $ v in B (0, R) $ ?

In other words, on a hexagonal grid, compare the modulation reduction of a vector going just above the top of the base hex (for this vector $ mod_L $ is far from $ mod_ ast $) to the one just past the middle of one of the sides of the base hex (for this vector they are close). What behavior dominates in high dimension?


I imagine that no, because I suspect that there are not enough regions of Voronoi adjacent to the base region for a mod reduction on network of a vector just slightly outside of the base region to approximately correspond to the reduction in length 2 of this vector.


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