On cap sets and the group-theoretic approach to matrix multiplication

Abstract

In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent ω of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific conjectures for how to obtain ω = 2. In this paper we rule out obtaining ω = 2 in this framework from abelian groups of bounded exponent. To do this we bound the size of tricolored sum-free sets in such groups, extending the breakthrough results of Croot, Lev, Pach, Ellenberg, and Gijswijt on cap sets. As a byproduct of our proof, we show that a variant of tensor rank due to Tao gives a quantitative understanding of the notion of unstable tensor from geometric invariant theory.

A tight bound for Green's arithmetic triangle removal lemma in vector spaces

“…Sets that contain no arithmetic progressions have also been studied over other abelian groups, see [6] and [20]. We expect these methods can be used in those settings as well to improve bounds on the corresponding removal lemma.…”

“…In particular, we have 1 c 2 = 5/3 − log 3 ≈ .0817 and [9] showed that their method can be used to greatly improve the upper bound on the cap set problem, and more generally to bound the size of a set in F n p with no 3-term arithmetic progressions. Further work by Blasiak-Church-Cohn-Grochow-NaslundSawin-Umans [6], and independently Alon, established that their proof also shows the following multicolored sum-free theorem over F n p . Later work by Kleinberg, Sawin, and Speyer [16] showed that there exists a computable exponent c p that is sharp.…”

A lower bound for the k‐multicolored sum‐free problem in Zmn

“…After Blasiak et al . established the upper bound, Kleinberg, Sawin, and Speyer proved Theorem for and any . Their proof uses a statement that had been formulated as a conjecture in an earlier version of their paper and was then proved by Pebody .…”

“…Soon after the preprint of Ellenberg and Gijswijt appeared, Blasiak, Church, Cohn, Grochow, Naslund, Sawin, and Umans and independently Alon noticed that the argument of Ellenberg and Gijswijt can also be used to obtain an upper bound on the size of ‐colored sum‐free sets in with (see the following definition). Definition Let be an abelian group and let .…”

“…Furthermore, the function tends to infinity when γ → 0, hence the minimum value Γ m,k is indeed attained for some 0 < γ m,k < 1 (one can show that there is a unique 0 < γ m,k < 1 where the minimum is attained, but this is not necessary for our purposes). Tao's slice rank method [35], together with the arguments from Blasiak et al [8], gives the following upper bound for the size of k-colored sum-free sets in Z n m for prime powers m. For the reader's convenience, we give a proof of Theorem 1.2 in Section 9 (see also [26,Theorem 4], which is a very similar theorem). Theorem 1.2.…”