Order-preserving Freiman isomorphisms

An order-preserving Freiman 2-isomorphism is a map $ϕ:X \rightarrow \mathbb{R}$ such that $ϕ(a) < ϕ(b)$ if and only if $a < b$ and $ϕ(a)+ϕ(b) = ϕ(c)+ϕ(d)$ if and only if $a+b=c+d$ for any $a,b,c,d \in X$. We show that for any $A \subseteq \mathbb{Z}$, if $|A+A| \le K|A|$, then there exists a subset $A' \subseteq A$ such that the following holds: $|A'| \gg_K |A|$ and there exists an order-preserving Freiman 2-isomorphism $ϕ: A' \rightarrow [-c|A|,c|A|] \cap \mathbb{Z}$ where $c$ depends only on $K$. Several applications are also presented.

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