Short-vector cancellation for flat tori

Let ${\Lambda }_1,{\Lambda }_2\subset {\mathbb{R}}^d$ be full-rank Euclidean lattices, and choose $\epsilon$ smaller than every nonzero vector length in either. The shortest vectors of ${\Lambda }_i\oplus \epsilon {\mathbb{Z}}^k$ are exactly the new coordinate vectors and their negatives. Any isometry between the associated flat tori lifts to an affine Euclidean isometry, whose orthogonal part maps the lattices. It must preserve the span of the shortest vectors, hence also its orthogonal complement, and therefore maps ${\Lambda }_1$ to ${\Lambda }_2$. Consequently a nonisometric pair remains nonisometric after adding sufficiently short equal circle factors. The spectrum of a flat torus shows that adding the same factors preserves isospectral manifolds.