 Solution

Let $H\le {\mathbb{F}}_p^n$ be a subspace with $|H|$ much larger than $K^2$, and choose linearly independent vectors $e_1,\dots ,e_{K-1}$ whose images are independent modulo $H$. SetThen $|A|=|H|+K-1\sim |H|$, while $A+A$ is the union of $H$, the $K-1$ disjoint cosets $e_i+H$, and at most $K^2$ exceptional sums $e_i+e_j$. HenceEvery subspace containing $A$ must contain $H$ and all $e_i$, so it has at least $p^{K-1}|H|\sim p^{K-1}|A|$ elements. Thus the exponential dependence on $K$ in the Freiman-Ruzsa theorem over a finite field cannot in general be replaced by a subexponential one.

Solved by gpt-5.6-sol high.