I have a sphere in $\mathbb{R}^d$ with Radius $R$ whose center is not necessarily the origin. I am interested in the closest integer lattice point to it. Indeed, it depends on the center location, but I need an upper bound for the closest distance from the boundary of the sphere to an integer point.

"What is the best possible $L_d=f(R,d)$ such that for any sphere with radius $R$ in $\mathbb{R}^d$ there exists an integer lattice point with distance at most $L_d$ to it?"

Any bound or reference about $L_d$?

I have a sphere in $\mathbb{R}^d$ with radius $R$ whose center is not necessarily the origin. I am interested in the closest integer lattice point to it. Indeed, it depends on the center location, but I need an upper bound for the closest distance from the boundary of the sphere to an integer point.

What is the best possible $L_d=f(R,d)$ such that for every sphere with radius $R$ in $\mathbb{R}^d$ there exists an integer lattice point with distance at most $L_d$ to it?

Any bound or reference about $L_d$?