Bull. Korean Math. Soc. 2024; 61(5): 1339-1367
Online first article September 25, 2024 Printed September 30, 2024
https://doi.org/10.4134/BKMS.b230631
Copyright © The Korean Mathematical Society.
Mohan, Ram Krishna Pandey
Indian Statistical Institute Delhi; Indian Institute of Technology Roorkee
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.
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