The Euclidean minimum M(K) of a number field K is an important numerical invariant that measures to what extent K is norm-Euclidean. It is the supremum of the Euclidean spectrum Spec(K) of K. The computability of M(K) as well as its isolatedness in Spec(K) has been much studied in computational number theory. In this talk, we will discuss how previous works by Lindenstrauss and Wang on topological rigidity of Z^r-actions by toral automorphisms can be applied to such studies. In particular, we will give an upper bound for the computational complexity of M(K) for non-CM fields of unit rank strictly greater than 1. For CM fields of unit rank strictly greater than 2, we show that M(K) is attained and isolated in Spec(K). Combined with Cerri's work, our result implies M(K) can be computed in finite time for all K of degree 7 or higher.

This is a joint work with Uri Shapira.