Extended inverse theorems for restricted sumset in integers
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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 $\left|4^{\wedge}A\right| = 4k-14$.
Abstract
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 $\left|4^{\wedge}A\right| = 4k-14$.
참고문헌 (16)
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