Let $\vec v$ be a vector in $\mathbb Z^3$. Then, what is the lower bound of $\|\vec v\|$? I think $\vec v$ is an integer vector, we can find a lower bound, but it is not easy for me to find it.
Actually, I tried to find a bound using a condition that $a^2 + b^2 + c^2 \ge 3 \sqrt[3] {a^2b^2c^2}$, but it looks so dirty.
Moreover, I tried to find some technique for lattice theory because of $\mathbb Z ^3$ is a lattice, but many books does not say it. Can we find a lower bound of norm of integer vector?