Direct and Inverse Problems for Restricted Signed Sumsets -- I
Raj Kumar Mistri, Nitesh Prajapati
TL;DR
We study the direct and inverse problems for the restricted $h$-fold signed sumset $h^{\wedge}_{\pm}A$ in additive abelian groups, focusing on $G = \mathbb{Z}$. The authors develop a suite of auxiliary lemmas and constructive techniques to derive tight lower bounds for $|h^{\wedge}_{\pm}A|$ and to characterize extremal sets. They prove the conjectures of Bhanja, Komatsu and Pandey for sets of positive integers, showing that equality in the extremal bound forces $A$ to be a dilate of the odd progression $\{1,3,\ldots,2k-1\}$; a follow-up work extends these results to nonnegative integers. The results advance understanding of restricted signed sumsets and inverse problems in additive combinatorics, with potential implications for related sumset problems and broader applications in number theory.
Abstract
Let $A=\{a_{1},\ldots,a_{k}\}$ be a nonempty finite subset of an additive abelian group $G$. For a positive integer $h$, the $h$-fold signed sumset of $A$, denoted by $h_{\pm}A$, is defined as $$h_{\pm}A=\left\lbrace \sum_{i=1}^{k} λ_{i} a_{i}: λ_{i} \in \{-h, \ldots, 0, \ldots, h\} \ \text{for} \ i= 1, 2, \ldots, k \ \text{and} \ \sum_{i=1}^{k} \left|λ_{i} \right| =h\right\rbrace,$$ and the restricted $h$-fold signed sumset of $A$, denoted by $h^{\wedge}_{\pm}A$, is defined as $$h^{\wedge}_{\pm}A=\left\lbrace \sum_{i=1}^{k} λ_{i} a_{i}: λ_{i} \in \left\lbrace -1, 0, 1\right\rbrace \ \text{for} \ i= 1, 2, \ldots, k \ \text{and} \ \sum_{i=1}^{k} \left|λ_{i} \right| = h\right\rbrace. $$ A direct problem for the sumset $h^{\wedge}_{\pm}A$ is to find the optimal size of $h^{\wedge}_{\pm}A$ in terms of $h$ and $|A|$. An inverse problem for this sumset is to determine the structure of the underlying set $A$ when the sumset $h^{\wedge}_{\pm}A$ has optimal size. While some results are known for the signed sumsets in finite abelian groups due to Bajnok and Matzke, not much is known for the restricted $h$-fold signed sumset $h^{\wedge}_{\pm}A$ even in the additive group of integers $\Bbb Z$. In case of $G = \Bbb Z$, Bhanja, Komatsu and Pandey studied these problems for the sumset $h^{\wedge}_{\pm}A$ for $h=2, 3$, and $k$, and conjectured the direct and inverse results for $h \geq 4$. In this paper, we prove these conjectures completely for the sets of positive integers. In a subsequent paper, we prove these conjectures for the sets of nonnegative integers.