Let $r_1, r_2, \dots, r_n > 0$ be positive real numbers and let
$$ E: \Big(\frac{x_1}{r_1}\Big)^2+\Big(\frac{x_2}{r_2}\Big)^2+\dots+\Big(\frac{x_n}{r_n}\Big)^2 = 1 $$
be the corresponding ellipsoid inside $\mathbb{R}^n$. Moreover, assume that there is no integer point on the ellipsoid $E$ (this holds, for example, when $r_i^2$ are $n$ algebraically independent numbers). Is there a formula for the distance between the ellipsoid $E$ and the integer lattice $\mathbb{Z}^n$?