Cyclic Freiman model of a small-doubling integer set ($|A^{\prime }|\ge |A|/9,A^{\prime }{\cong }_{8}B\subseteq \mathbb{Z}/q\mathbb{Z}$)

For $|A+A|\le K|A|$, one can take $q=8|8A-8A|\le 8K^{16}|A|$. Embed in a much larger prime cyclic group and choose a dilation avoiding the residues of all nonzero multiples of $q$ of absolute value below that prime for every nonzero eightfold difference. A union bound ensures such a dilation. Retain a most-populated ninth-interval of representatives. Eight-term sums then have range shorter than the prime, and reduction modulo $q$ both preserves and reflects eight-term additive relations.