= Cyclic Freiman model of a small-doubling integer set
{title2=$|A'|\ge|A|/9,\quad A'\cong_8 B\subseteq\mathbb Z/q\mathbb Z$}
For $|A+A|\le K|A|$, one can take $q=8|8A-8A|\le8K^{16}|A|$. 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.
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