Let $r(n):=r_3(\mathbb{F}_3^n)=\max\{|A|: A \subset \mathbb{F}_3^n, \ A \text{ is 3-AP-free}\}$.

Edel proved that $r(n)\geq 2.21^n$ for sufficiently large $n$. His proof is by giving a construction of a cap-set $A$ in $F_3^{62}$. Then observing that $A^k \subset F_3^{62k}$ is also a cap set, that is, $$r(62k)\geq |A^k|=|A|^k.$$

Is this the best known lower bound? Are there other known approaches to this problem other than construction in low dimension and then using this product argument?

Is this product argument expected to be the best we could expect? That is, do we hope to construct an $A$ such that this argument is tight?

I'd appreciate any references or answers to some of these questions.

Any help on the tags or on how to better ask these questions would be nice also.