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 -…
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