{"id":"hexagon/2610.00559","pdf":"https://hexagonmath.org/pdf/2610.00559v1","url":"https://hexagonmath.org/abs/2610.00559v1","tags":["msc:11H06","msc:11H55","msc:11E16","msc:53C22","msc:58J50","lattice geometry; flat tori"],"score":0,"title":"Finite Truncated Lattice-Length Data on Three-Dimensional Flat Tori","domain":"math.MG","source":"hexagon","authors":["Changxia Lin"],"reviews":0,"subject":"math.MG","summary":"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 -…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-26","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}