Some inverse results of sumsets

TL;DR

This paper investigates the inverse problem in additive number theory, characterizing finite integer sets based on extremal cardinalities of their sumsets, extending known results about arithmetic progressions.

Contribution

It provides new inverse results for finite sets of integers with extremal sumset cardinalities, broadening understanding beyond arithmetic progressions.

Findings

Characterization of sets with extremal sumset sizes

Extension of classical inverse sumset results

Identification of non-arithmetic progression structures

Abstract

Let $h\geq 2$ and $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a finite set of integers. It is well-known that $\left|hA\right|=hk-h+1$ if and only if $A$ is a $k$-term arithmetic progression. In this paper, we give some nontrivial inverse results of the sets $A$ with some extrema the cardinalities of $hA$.