For high enough dimension $n$ there are lattices $L_n$ in $\mathbb{R}^n$ whose Voronoi partition's base regions encompass all but a negligible proportion of a $(1-\varepsilon)$-ball, and also nearly contain a $(1+\varepsilon)$-ball to all but a negligible proportion.[1] How much can such base regions behave like a ball in other ways?

One can define a "ball mod" as $mod_\ast(v)=(v/\|v\|)\cdot [\|v\| - \ \operatorname{round}(\|v\|/2)]$, which just reduces $v$ by lengths of 2 until it is in the unit ball at the origin.

One can also define "round" and "mod" operations over space up to a lattice $L$ and its Voronoi partition as: $R_L(v)=\arg\min_{\ell\in L}\|v-\ell\|$ and $mod_L(v)=v-R_L(v).$

Fixing radius $R>0$, for high enough dimension $n$, then does a lattice $L$ of the sort described above have that: $\|mod_\ast(v)-\operatorname{mod}_L(v)\|<\varepsilon$ for most $v\in B(0,R)$ ?