= Cardinality-controlled cyclic Freiman model
{title2=$|A'|\ge|A|/(k+1),\quad N=4|kA-kA|+1$}
A nonempty finite integer set has a subset of the displayed size <Freiman s-isomorphic> to a subset of $\mathbb Z_N$. Embed in a sufficiently large auxiliary prime modulus and randomly dilate. Avoid the residues of every nonzero multiple of $N$ of absolute value less than that prime for all nonzero members of $kA-kA$; a <union bound> gives a successful dilation. Retain a most-populated one of $k+1$ short intervals of representatives. Its $k$-term sum differences have absolute value less than the auxiliary prime, so reduction modulo $N$ both preserves and reflects relations. Under $|kA-kA|\le C|A|$, this gives $N\le4C|A|+1$.
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