The next gap in the subrank of 3-tensors

TL;DR

This paper identifies the next possible value gap in the asymptotic subrank and slice rank of 3-tensors, extending previous work on the discrete set of these tensor parameters.

Contribution

It precisely determines the next gap in the set of possible asymptotic subrank and slice rank values for 3-tensors, building on prior discrete value classifications.

Findings

The asymptotic subrank and slice rank can be exactly 1, 1.88, 2, or at least 2.68.

The set of possible values of these parameters is discrete.

The paper extends the understanding of tensor rank gaps.

Abstract

Recent works of Costa-Dalai, Christandl-Gesmundo-Zuiddam, Blatter-Draisma-Rupniewski, and Bri\"et-Christandl-Leigh-Shpilka-Zuiddam have investigated notions of discreteness and gaps in the possible values that asymptotic tensor ranks can take. In particular, it was shown that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, or at least 2 (over any field), and that the set of possible values of these parameters is discrete (in several regimes). We determine exactly the next gap, showing that the asymptotic subrank and asymptotic slice rank of any nonzero 3-tensor is equal to 1, equal to 1.88, equal to 2, or at least 2.68.