Tensor parameters that are amortized or regularized over large tensor powers, often called "asymptotic" tensor parameters, play a central role in several areas including algebraic complexity theory (constructing fast matrix multiplication algorithms), quantum information (entanglement cost and distillable entanglement), and additive combinatorics (bounds on cap sets, sunflower-free sets, etc.). Examples are the asymptotic tensor rank, asymptotic slice rank and asymptotic subrank. Recent works (Costa-Dalai, Blatter-Draisma-Rupniewski, Christandl-Gesmundo-Zuiddam) have investigated notions of discreteness (no accumulation points) or "gaps" in the values of such tensor parameters. We prove a general discreteness theorem for asymptotic tensor parameters of order-three tensors and use this to prove that (1) over any finite field (and in fact any finite set of coefficients in any field), the asymptotic subrank and the asymptotic slice rank have no accumulation points, and (2) over the complex numbers, the asymptotic slice rank has no accumulation points. Central to our approach are two new general lower bounds on the asymptotic subrank of tensors, which measures how much a tensor can be diagonalized. The first lower bound says that the asymptotic subrank of any concise three-tensor is at least the cube-root of the smallest dimension. The second lower bound says that any concise three-tensor that is "narrow enough" (has one dimension much smaller than the other two) has maximal asymptotic subrank. Our proofs rely on new lower bounds on the maximum rank in matrix subspaces that are obtained by slicing a three-tensor in the three different directions. We prove that for any concise tensor, the product of any two such maximum ranks must be large, and as a consequence there are always two distinct directions with large max-rank.

翻译：张量参数经大张量幂的摊销或正则化后常被称为“渐近”张量参数，在代数复杂性理论（构建快速矩阵乘法算法）、量子信息（纠缠代价与可蒸馏纠缠）及加性组合学（帽集界限、无向日葵集等）等多个领域发挥核心作用。典型例子包括渐近张量秩、渐近切片秩及渐近子秩。近期研究（Costa-Dalai、Blatter-Draisma-Rupniewski、Christandl-Gesmundo-Zuiddam）探讨了此类张量参数值的离散性（无聚点）或“间隙”概念。本文证明了三阶张量渐近参数的一般离散性定理，并由此证明：(1) 在任意有限域（实际上任意域中的任意有限系数集）上，渐近子秩与渐近切片秩无聚点；(2) 在复数域上，渐近切片秩无聚点。我们方法的核心在于两个关于张量渐近子秩（度量张量可对角化程度）的新的一般下界：第一个下界表明任意简洁三阶张量的渐近子秩至少为最小维度的立方根；第二个下界指出任何“足够窄”（某一维度远小于另两个维度）的简洁三阶张量具有最大渐近子秩。我们的证明依赖于对三阶张量沿三个不同方向切片所得矩阵子空间最大秩的新下界估计。我们证明，对任意简洁张量，任意两个此类最大秩的乘积必须足够大，因此必然存在两个不同方向具有大最大秩。