On Freiman's 3k-4 theorem

TL;DR

This paper introduces a new structural concept for finite integer subsets and extends Freiman's 3k-4 theorem to groups of the form Z×G, enhancing understanding in additive number theory.

Contribution

It proposes a novel structure for finite integer sets and generalizes Freiman's 3k-4 theorem to abelian groups of the form Z×G.

Findings

Introduction of a new structural concept for integer subsets

Establishment of an analog of Freiman's 3k-4 theorem for Z×G groups

Provides criteria for subsets to possess this new structure

Abstract

One of the many theorems Freiman proved, in the second half of the twentieth century, in the subject which later came to be known as "structure theory of set addition", was 'Freiman's $3k-4$ theorem' for subsets of $\Z$. In this article we introduce concept of a new `structure' on finite subsets of integers. Sets with this structure are quite useful in additive number theory in some contexts. Also we give some criterion for subsets to posses this structure. Then this is used to establish an analog of Freiman's $3k-4$ theorem for the groups $\Z \times G,$ where $G$ is any abelian group.