Bulletin of the
Korean Mathematical Society BKMS

Let $h$ and $k$ be positive integers such that $h\leq k$. Let $A = \{a_{0}, a_{1}, \ldots, a_{k-1}\}$ be a nonempty finite set of $k$ integers.  The \textit{h-fold sumset}, denoted by $hA$, is a set of integers that can be expressed as a sum of $h$ elements (not necessarily distinct) of $A$. The \textit{restricted h-fold sumset}, denoted by $h^{\wedge}A$, is a set of integers that can be expressed as a sum of $h$ distinct elements of $A$.   The characterization of the underlying set for small deviation from the minimum size of the sumset is called an \textit{extended inverse problem}. Freiman studied such a problem and proved a theorem for $2A$, which is  known as \textit{Freiman's $3k-4$ theorem}. Very recently, Tang and Xing, and Mohan and Pandey studied some more extended inverse problems for the sumset $hA$, where $h\geq 2$.  In this article, we prove some extended inverse theorems for sumsets $2^{\wedge}A$, $3^{\wedge}A$ and $4^{\wedge}A$. In particular, we  classify the  set(s) $A$ for which $\left|2^{\wedge}A\right| =2k-2$, $\left|2^{\wedge}A\right| =2k-1$, and $\left|2^{\wedge}A\right| =2k$.  Furthermore, we also classify  set(s) $A$ when  $\left|3^{\wedge}A\right| = 3k-7$, $\left|3^{\wedge}A\right| = 3k-6$, and $\left|4^{\wedge}A\right| = 4k-14$.

Keywords: Sumset, restricted sumset, inverse problem, extended inverse problem

MSC numbers: 11P70, 11B75, 11B13, 11B25

Supported by: This work was done when the first author was at the Department of Mathematics, Indian Institute of Technology Roorkee, India, and the first author was financially supported by Council of Scientific and Industrial Research, Grant No. 09/143(0925)/2018-EMR-I.