I would like to cover the $n-1$ dimensional hyperplane $H:=\{x\in \mathbb R^n\mid \langle v, x\rangle =0\}$ by balls around $(I-\frac{vv^T}{\langle v,v\rangle})\mathbb Z^n$, where $v\in \mathbb Q^n$. By cover I mean, that I like to find a small number $r>0$ such that $$H\subset \left(I-\frac{vv^T}{\langle v,v\rangle}\right)\mathbb Z^n+B_r(\mathbf 0).$$
My thoughts so far:
- If one of the entries of $v$ is not a rational number, then I suppose, that $(I-\frac{vv^T}{\langle v,v\rangle})\mathbb Z^n$ is not a discrete set. Maybe for some choices the covering radius is zero, but I am not interested in these cases. I'd like $v$ to be in $\mathbb Q^n$.
- The matrix $\left(I-\frac{vv^T}{\langle v,v\rangle}\right)$ is symmetric and has the eigenvalues 1 and 0 with the eigenspaces $H$ respectively $H^\bot=\operatorname{span}\{v\}$.
If $v=(1,1,\ldots,1,1)^T$, then I can show that the radius is less than $\frac {\sqrt n}{2\sqrt 3}$. But I like to find a vector $v$ such that the radius is less than $\frac{\sqrt n}{3\sqrt 2}$. Is it possible?