Let $\mathbf u\in\mathbb Z^d$ be a primitive vector (i.e. $\gcd(u_i)=1$) and let $\Pi_{\mathbf u^\perp}$ be the orthogonal projection perpendicular to $\mathbf u$. I want to understand the geometry of the $(d-1)$-dimensional lattice, $L=\{\Pi_{\mathbf u^\perp}\mathbf n\colon \mathbf n\in\mathbb Z^d\}$. Specifically:
I want to relate $\Delta:=\max_{x\in\mathbb R^d}d(\Pi_{\mathbf u^\perp}x,L)$ to the vector $\mathbf u$.
The kind of answer I'd really like might give an upper bound for $\Delta$ in terms of $\min\{\langle \mathbf v,\mathbf v\rangle\colon \mathbf v\in\mathbb Z^d\setminus\{0\}; \langle\mathbf u,\mathbf v\rangle=0\}$). Can anyone point me in the right direction?