Order-preserving Freiman isomorphisms

Abstract

An order-preserving Freiman 2-isomorphism is a map $\varphi :X\to \mathbb{R}$ such that $\varphi (a)<\varphi (b)$ if and only if $a<b$ and $\varphi (a)+\varphi (b)=\varphi (c)+\varphi (d)$ 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 $A^{\prime }\subseteq A$ such that the following holds: $|A^{\prime }|{\gg }_K|A|$ and there exists an order-preserving Freiman 2-isomorphism $\varphi :A^{\prime }\to [-c|A|,c|A|]\cap \mathbb{Z}$ where $c$ depends only on $K$. Several applications are also presented.

14 pages

References in corpus (1)

• On distinct consecutive $r$-differences