Order-preserving Freiman isomorphisms

TL;DR

This paper proves that sets with small sumsets in integers can be embedded into bounded intervals via order-preserving Freiman isomorphisms, with applications in additive combinatorics.

Contribution

It establishes the existence of large subsets with order-preserving Freiman isomorphisms into bounded intervals for sets with small doubling.

Findings

Existence of large subsets with structured embeddings

Bounded interval embeddings depend only on doubling constant

Applications in additive combinatorics and related fields

Abstract

An order-preserving Freiman 2-isomorphism is a map $\phi:X \rightarrow \mathbb{R}$ such that $\phi(a) < \phi(b)$ if and only if $a < b$ and $\phi(a)+\phi(b) = \phi(c)+\phi(d)$ 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 $\phi: A' \rightarrow [-c|A|,c|A|] \cap \mathbb{Z}$ where $c$ depends only on $K$. Several applications are also presented.