On Doubling and Volume: Chains

TL;DR

This paper proves Freiman's conjecture on the maximum volume of sets with small doubling property for a class called chains, advancing understanding of additive combinatorics and set structure.

Contribution

It confirms Freiman's conjecture for chains, a significant class of sets, providing explicit bounds on their volume in the context of small doubling.

Findings

Proof of Freiman's conjecture for chains

Explicit bounds on set volume in small doubling case

Enhanced understanding of set structure in additive combinatorics

Abstract

The well--known Freiman--Ruzsa Theorem provides a structural description of a set $A$ of integers with $|2A|\le c|A|$ as a subset of a $d$--dimensional arithmetic progression $P$ with $|P|\le c'|A|$, where $d$ and $c'$ depend only on $c$. The estimation of the constants $d$ and $c'$ involved in the statement has been the object of intense research. Freiman conjectured in 2008 a formula for the largest volume of such a set. In this paper we prove the conjecture for a general class of sets called chains.