A gap in the slice rank of k-tensors

Abstract

The slice-rank method, introduced by Tao as a symmetrized version of the polynomial method of Croot, Lev and Pach and Ellenberg and Gijswijt, has proved to be a useful tool in a variety of combinatorial problems. Explicit tensors have been introduced in different contexts but little is known about the limitations of the method. In this paper, building upon a method presented by Tao and Sawin, it is proved that the asymptotic slice rank of any k-tensor in any field is either 1 or at least k/(k−1)(k−1)/k. This provides evidence that straight-forward application of the method cannot give useful results in certain problems for which non-trivial exponential bounds are already known. An example, actually a motivation for starting this work, is the problem of bounding the size of sets of trifferent sequences, which constitutes a long-standing open problem in information theory and in theoretical computer science.