Isolation of Euclidean minima in higher unit rank

Prove that the norm Euclidean minimum is isolated in number fields of unit rank greater than $1$.

Background

The first Euclidean minimum need not be isolated in imaginary quadratic fields. In contrast, Barnes and Swinnerton-Dyer proposed that isolation should hold in number fields with unit rank greater than one; the paper records this as a conjectural structural property of Euclidean minima.

References

Barnes and Swinnerton-Dyer, on the other hand, have conjectured that it is always isolated in number fields of unit rank $>1$.

— Euclidean Rings  (2608.23216 - Lemmermeyer, 24 Aug 2026) in Section 0, discussion following the definition of the second Euclidean minimum

For number fields of arbitrary signature not even an analogous conjecture is known.

— Euclidean Rings  (2608.23216 - Lemmermeyer, 24 Aug 2026) in Remarks on Section 2

The difficulty in a generalization of this theorem to fields of arbitrary unit rank $\ge 1$ lies solely in the construction of a $y \in K$ with the property that the finitely many $y_u$ lie near the corresponding $x_u$. I have not, however, yet succeeded in carrying out such a construction.

— Euclidean Rings  (2608.23216 - Lemmermeyer, 24 Aug 2026) in Section 2, immediately after the proof of (2.12)