= Removal of an exponential prefactor by Cartesian powers
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 $|A|^k\leq K\theta^{nk}$. Taking $k$th roots and letting $k$ tend to infinity proves $|A|\leq\theta^n$. More generally this argument applies to any class of finite objects closed under products with multiplicative size and additive dimension.
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