While connected arithmetic discrete lines are entirely characterized, only partial results exist for arithmetic discrete hyperplanes in any dimension. In the present paper, we focus on $0$-connected rational arithmetic discrete planes in $\Z^3$. Thanks to an arithmetic reduction on the parameters of a given integer vector $\vect{n}$, we provide an algorithm which computes the thickness of the thinnest $0$-connected arithmetic plane with normal vector $\vect{n}$.