Cardinality-controlled cyclic Freiman model ($|A^{\prime }|\ge |A|/(k+1),N=4|kA-kA|+1$)

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 $N\le 4C|A|+1$.