Cartesian powers of cap sets

Every Cartesian product of cap sets is a cap set. In characteristic three, $x+y+z=0$ with any two equal forces all three equal. A zero-sum triple in a Cartesian product of cap sets is therefore diagonal in every coordinate block, hence diagonal overall.

Suppose every cap set in ${\mathbb{F}}_3^n$ has cardinality at most $K{\theta }^n$, with $K$ independent of $n$. Applying this to the $k$-fold Cartesian product of a fixed cap set $A$ gives $|A{|}^k\le K{\theta }^{nk}$. Taking $k$th roots and letting $k$ tend to infinity proves $|A|\le {\theta }^n$. More generally this argument applies to any class of finite objects closed under products with multiplicative size and additive dimension.