Freiman s-isomorphism (Freiman $s$-isomorphism)

A Freiman $s$-isomorphism is a bijection whose forward and inverse maps are both Freiman s-homomorphisms.

An eight-term Freiman isomorphism extends injectively to the two-sum/two-difference set by sending its representations to the corresponding integer differences. Independence of representation uses four-term relations, and additivity whenever $y,z,y+z$ lie in that difference set uses six-term relations. Thus a proper symmetric progression in the modeled difference set lifts, one coordinate step at a time, to a proper integer progression of the same rank and cardinality.

A Freiman 2-isomorphism is a bijection between subsets of abelian groups preserving additive quadruple relations in both directions. It need not extend to an ambient group homomorphism. Multiplication by a nonzero scalar in a finite field and translation are examples; reduction of an integer set modulo a prime is an example only when it creates no new additive relations.

If a Freiman 2-isomorphism into ${\mathbb{F}}_2^m$ uses minimal $m$, every nonzero vector lies in the fourfold sumset. Otherwise quotienting by the line it spans would preserve all additive quadruple relations and would still be injective, contradicting minimality. Since zero also belongs to the fourfold sumset, that sumset is the full model group. Plünnecke-Ruzsa inequality then bounds its size by $K^4|A|$. A finite subset of the direct sum of countably many binary fields has a finite-dimensional model to start with.