Using the slice rank for finding upper bounds on the size of cap sets

Using the slice rank for finding upper bounds on the size of cap sets

Borst, Sander (TU Delft Electrical Engineering, Mathematics and Computer Science)

Gijswijt, Dion (mentor)

Delft University of Technology

2018-01-15

The cap set problem consists of finding the maximum size cap sets, i.e. sets without a 3-term arithmetic progression in F₃. In this thesis several known results on the behavior of this number as n → ∞ are presented. In particular we discuss a reformulation by Terence Tao and Will Sawin of a proof found by Dion Gijswijt and Jordan Ellenberg. It uses the slice rank, a rank that is defined for elements of tensor products, to give upper bounds on the size of the cap sets. In this report we will explain the slice rank and how it is related to the size of cap sets. We will also explore whether the slice rank might be used for bounding the size of arithmetic progression-free sets in F_q for q ≠ 3. We show that we can not use the slice rank to give a non-trivial upper bound on the size of n-term progression-free sets for n ≥ 7. This was already known for n ≥ 8.

cap set
slice rank
arithmetic progression

http://resolver.tudelft.nl/uuid:a619c626-8a7e-45d5-90e4-1ff4cfa5268b

Student theses

bachelor thesis

© 2018 Sander Borst