# 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