The Closest Vector Problem for the lattice of algebraic integers

Any number field comes with a natural inner product as in the theory of the geometry of numbers, so that any order becomes a lattice.
We extend the definition of the inner product to $\overline{\mathbb{Q}}$, the algebraic closure of the rationals, and consider its maximal order $\overline{\mathbb{Z}}$, which has infinite rank, as an intrinsically interesting `lattice'.
We will compute several lattice invariants and attempt to solve the Closest Vector Problem through proofs inspired by capacity theory.
Adjacent to CVP is the problem of finding the Voronoi-relevant vectors, and we pose the challenge to compute all such vectors of degree 3.