Bachelor-eindpresentatie: Cap sets

15 januari 2018 13:30 - Locatie: Gebouw 36: Faculteit EWI, Timmanzaal (LB 01.170) - Door: Communicatie EWI

Vandaag presenteert Sander Borst zijn bacheloreindproject: Using slice rank for finding upper bounds on the size of cap sets

abstract

The cap set problem consists of finding the maximum size cap sets, i.e. sets without a 3-term arithmetic progression in F_3. 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.