math.MG — Metric Geometry
Finite Truncated Lattice-Length Data on Three-Dimensional Flat Tori
We study what finite truncations of the lattice-length multiset can determine about a three-dimensional flat torus. For a full-rank lattice \Lambda \subset \mathbb{R}^3, let \lambda_1(\Lambda) \leq \lambda_2(\Lambda) \leq \lambda_3(\Lambda) be its successive minima, and let \mathcal{L}_T(\Lambda) record the lengths of all nonzero lattice vectors of norm at most T, counted with vector multiplicity. First, on the class defined by \lambda_1 = a and \lambda_2 > T, we prove an exact volume classification: the set of possible covolumes is (F(a,T), \infty), where F(a,T) = \frac{aT}{2}\sqrt{3T^2 - a^2}. The lower bound uses Oppenheim's classical determinant-minimization theorem for Minkowski-reduced ternary forms; an explicit perturbed basis proves every larger volume is attained. Second, we give a pair of positive-definite Gram matrices with equal diagonal entries and matching two-coordinate data but different determinants. This yields a quantitative obstruction to uniform finite identification and gives the lower bound \sqrt{3} for the worst-case identification-radius constant. Schiemann's finite theta-series determination theorem supplies the complementary upper bound \sqrt{7/2} after conversion from squared lengths to radii. We also record a cubic sign-product identity for the eight signed basis vectors and explain its limitations. All contributions are separated from classical inputs, and the possible optimality of the constant \sqrt{3} and the broader topological question are left open.