Direct and Inverse Problems for Restricted Signed Sumsets -- II
Raj Kumar Mistri, Nitesh Prajapati
TL;DR
This work resolves direct and inverse problems for the restricted $h$-fold signed sumset $h^{\wedge}_{\pm}A$ in additive groups, focusing on nonnegative integers. It proves a sharp direct lower bound $|h^{\wedge}_{\pm}A| \ge 2hk - h^2 - h + 1$ for $3 \le h \le k-1$ and $k \ge 5$, and provides a complete inverse characterization: equality forces $A$ to be a dilate of an arithmetic progression, specifically $A = a_2 \ast [0,k-1]$, with explicit exceptional cases when $h=4$ and $k=5$ allowing $A$ to take the forms $a_2 \ast [0,4]$ or $a_2 \ast \{0,1,2,4,6\}$. A parallel, sharp description is given for the bound $|h^{\wedge}_{\pm}A| = 2hk - h(h+1) + 1$, again yielding that extremal $A$ must be of the form $A = a_2 \ast [0,k-1]$ (plus the same small-exception caveats). The results fully settle the conjectures for nonnegative integers and extend the understanding of restricted signed sumsets, employing a suite of auxiliary lemmas that analyze parity structure, disjoint sumsets, and sumset decompositions.
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 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 restricted $h$-fold signed sumset 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 has optimal size. While the signed sumsets (which is defined differently compared to the restricted signed sumset) in finite abelian groups has been investigated by Bajnok and Matzke, the restricted $h$-fold signed sumset $h^{\wedge}_{\pm}A$ is not well studied even in the additive group of integers $\Bbb Z$. Bhanja, Komatsu and Pandey studied these problems for the restricted $h$-fold signed sumset for $h=2, 3$, and $k$, and conjectured some direct and inverse results for $h \geq 4$. In a recent paper, Mistri and Prajapati proved these conjectures completely for the set of positive integers. In this paper, we prove these conjectures for the set of nonnegative integers, which settles all the conjectures completely.