After the cap-set problem, and some properties of the slice rank

Abstract: The infamous cap-set problem asks for the size of the largest subset $A \subset \mathbb{F}_3^n$ not containing any solutions to the equation $x+y+z=0$ aside from the trivial solutions $x=y=z$. A proof that that size is bounded above by $C^n$ for some $C<3$, which arose in 2016 in two breakthrough papers by Croot-Lev-Pach and by Ellenberg and Gijswijt (both published in the Annals of Mathematics), was later reformulated by Tao in a more symmetric way, leading to the definition of a new notion of rank on tensors called the slice rank.

Since then, the slice rank has been studied further, and the resulting properties have often found related number-theoretic applications. To take the earliest and perhaps simplest example, a key component of the argument in the proof of the original cap-set problem itself is that the slice rank of a “diagonal” tensor is equal to its number of non-zero entries, mirroring the analogous property of matrix rank.

After reviewing some more such applications by other mathematicians, we will present some results concerning other basic properties of the slice rank, and in particular the ideas behind some of their simpler proofs in the special case where the support of the tensor is contained in an antichain: there, as established by Sawin and Tao, the slice rank of the tensor is equal to the smallest number of slices that suffice to cover its support. If time allows then we will also discuss how the proofs in this special case illuminate to some extent the proofs in the general case.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)