Detailed Structure for Freiman's 3k-3 Theorem

TL;DR

This paper characterizes the structure of finite integer sets with sumsets of size exactly 3|A|-3, revealing specific configurations including bi-arithmetic progressions and Freiman isomorphisms.

Contribution

It provides a detailed structural classification for sets with minimal sumset size 3|A|-3, extending Freiman's 3k-3 theorem with explicit set descriptions.

Findings

Sets are either bi-arithmetic progressions or contain large arithmetic progressions.

Identifies specific Freiman isomorphic structures for sets with minimal sumset size.

Provides conditions for sets to be structured as unions of dense, anti-symmetric subsets.

Abstract

Let A be a finite set of integers. We prove that if |A| is at least 2 and |A+A| is 3|A|-3, then one of the following is true: 1. A is a bi-arithmetic progression; 2. A+A contains an arithmetic progression of length 2|A|-1; 3. |A| is 6 and A is Freiman isomorphic to the set {(0,0),(0,1),(0,2),(1,0),(1,1),(2,0)}; 4. A is Freiman isomorphic to a set in either the form of {0,2,...,2k} union B union {n} for some non-negative integer k at most n/2 -2 or the form of {0} union C union D union {n}, where n=2|A|-2, B is left dense in [2k,n-1], C is right dense in [1,u] for some u in [4,n-6], D is left dense in [u+2,n-1], B,C,D are anti-symmetric and additively minimal in the correspondent host intervals.