arXiv subjects

Raj Kumar Mistri

Publications and source records attributed to Raj Kumar Mistri.

8 recordsLinked to original sources

arXiv2026

Signed sumsets and restricted signed sumsets in groups and fields

Let $A = \{a_1, \ldots, a_k\}$ be a nonempty finite subset of an additive abelian group $G$. For a nonnegative integer $h$, the \emph{$h$-fold signed sumset} of $A$, denoted by $h_{\pm} A$, is defined by $$ h_{\pm} A = \Biggl\{\sum_{i = 1}^{k} \lambda_i a_i : \lambda_i \in \{-h, \ldots, h\}, \ \sum_{i = 1}^{k} |\lambda_i| = h \Biggr\}, $$ and the \emph{restricted $h$-fold signed sumset}, denoted by $h_{\pm}^\wedge A$, is defined by $$ h_{\pm}^\wedge A = \Biggl\{\sum_{i = 1}^{k} \lambda_i a_i : \lambda_i \in \{-1, 0, 1\}, \ \sum_{i = 1}^{k} |\lambda_i| = h \Biggr\}. $$ We study direct and inverse problems for these signed sumsets, namely determining extremal bounds for their sizes and characterizing the structure of sets $A$ attaining these bounds. While such problems have been extensively studied and resolved in the additive group of integers, comparatively little is known in general abelian groups, especially for restricted signed sumsets. In this paper, we investigate the signed sumset $h_{\pm} A$ in arbitrary (not necessarily finite) abelian groups under the condition $A \cap (-A) \neq \varnothing$. We further analyze both $h_{\pm} A$ and $h_{\pm}^\wedge A$ when $A \cap (-A)$ has a prescribed size. These results are extended to generalized signed sumsets $H_{\pm} A = \bigcup_{h \in H} h_{\pm} A$, where $H$ is a finite set of nonnegative integers, with particular attention to $[0,h]_{\pm} A$. Furthermore, using the polynomial method, we establish nontrivial lower bounds for $|h_{\pm}^\wedge A|$ in arbitrary fields. In addition, for $h = 2, 3, 4$, we derive lower bounds for $|h_{\pm} A|$ in arbitrary fields under the condition $A \cap (-A) = \varnothing$.

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arXiv2026

Restricted set addition in finite abelian groups

Let $A$ be a nonempty subset of finite abelian group $G$ of order $n$. For an integer $h \geq 2$, the restricted $h$-fold sumset $h^\wedge A$ is the set of all sums of $h$ distinct elements of $A$. It is known that if $G$ is a group of order $n$ and $A$ is a subset of $G$ such that $|A|$ is close to $\frac{n}{2}$, then $h^{\wedge}A = G$ under some conditions on $h$ and $n$. The constant $\frac{1}{2}$ is optimal for groups of even order but not for groups of odd order. For an integer $h \geq 4$, let $\alpha_h$ be the unique positive root of the polynomial $3^{h - 2} x^{h - 1} + x - 1$. In this paper, we show that for any $\alpha > \alpha_h$, there exists a positive integer $M_h(\alpha)$, which is determined precisely, such that for all $n > M_h(\alpha)$ with $n$ odd, if $A$ is a subset of a finite abelian group $G$ of order $n$ and if $|A| \geq \alpha n$, then $h^{\wedge} A = G$. Moreover, $\alpha_h > \alpha_{h + 1}$ for $h \geq 4$ and $\alpha_h$ approaches $\frac{1}{3}$ as $h$ increases, and the constant $\frac{1}{3}$ is optimal when the smallest prime dividing $n$ is $3$. This result extends a theorem of Tang and Wei on $4^{\wedge}A$ in the cyclic group $\mathbb{Z}_n$ to $h^{\wedge}A$ for every $h \geq 4$, and to arbitrary finite abelian groups.

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arXiv2026

Structure of sets with small product sets in torsion-free groups, cyclic groups of prime orders and abelian groups

Let $\ell$ and $m$ be positive integers with $\ell \leq m$, and let $\mathcal{A} = (A_1, \ldots, A_m)$ be a finite sequence of finite subsets of a group $G$ (not necessarily abelian), written multiplicatively. The {\it generalized product set} $\Pi^{\ell}(\mathcal{A})$ is the set of all elements of $G$ which can be represented as a product of exactly $\ell$ elements from $\ell$ distinct sets from $\mathcal{A}$ taken in any order. DeVos, Goddyn and Mohar obtained the nontrivial lower bound for the size of this product set when $G$ is abelian. The DeVos-Goddyn-Mohar Theorem is a fundamental result in additive combinatorics which unifies various results from zero-sum combinatorics and has connections with subsequence sums and sumsets. In this paper, we obtain an optimal lower bound for the size of generalized product set ${\Pi}^{\ell}(\mathcal{A})$ in torsion-free groups (not necessarily abelian), and characterize the structure of underlying sets in the sequence $\mathcal{A} = (A_1, \ldots, A_m)$ for which ${\Pi}^{\ell}(\mathcal{A})$ achieves the optimal lower bound. By slightly modifying the arguments of the proofs in the case of torsion-free groups, we derive such inverse theorems in cyclic groups of prime orders also. Our proof of these result also yields a new proof of DeVos-Goddyn-Mohar Theorem in $\mathbb{Z}_p$. Moreover, we extend these inverse results to arbitrary abelian groups. Furthermore, as an application, we generalize a theorem for subsequence sums due to Hamidoune in torsion-free groups, and obtain several other results for subsequence sums in arbitrary groups.

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arXiv2025

The maximum size of sumsets in finite cyclic groups

Let $A$ be a nonempty finite subset of an additive abelian group $G$. Given a nonnegative integer $h$, the $h$-fold sumset $hA$ is the set of all sums of $h$ elements of $A$, and the restricted $h$-fold sumset $h^\wedge A$ is the set of all sums of $h$ distinct elements of $A$. The union of restricted sumsets $s^\wedge A$, where $s=0, 1, \ldots, h$, is denoted by $[0, h]^\wedge A$. For fixed positive integers $m$ and $h$, the maximum size of the sumset $hA$ of a set $A \subseteq G$ with $m$ elements is denoted by $\nu(G, m, h)$. In other words, $\nu(G, m, h) = \max\{|hA| : A \subseteq G, |A|= m\}$. Analogous quantities can be defined for the sumsets $h^\wedge A$ and $[0, h]^\wedge A$. Optimal upper bounds are known for these quantities. If $G$ is a finite cyclic group of order $n$, then each of these quantities agrees with the optimal upper bound, except in many cases. Bajnok posed the problem of determining all positive integers $n$, $m$, and $h$ for which the value of the function $f(n, m, h)$ is strictly less than the optimal upper bound. He posed similar problems for quantities related to the sumsets $h^\wedge A$ and $[0, h]^\wedge A$. We prove that, for any positive integer $h$, there are infinitely many positive integers $m$ and $n$ such that $\nu(\mathbb{Z}_n, m, h)$ is strictly less than the optimal upper bound. We also prove similar results for quantities related to the sumsets $h^\wedge A$ and $[0, h]^\wedge A$ also. These results provide the partial solutions to the problems posed by Bajnok.

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arXiv2025

Direct and Inverse Problems for Restricted Signed Sumsets -- II

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 $$h^{\wedge}_{\pm}A = \left\lbrace \sum_{i=1}^{k} \lambda_{i} a_{i}: \lambda_{i} \in \left\lbrace -1, 0, 1\right\rbrace \ \text{for} \ i= 1, 2, \ldots, k \ \text{and} \ \sum_{i=1}^{k} \left|\lambda_{i} \right| =h\right\rbrace. $$ 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.

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arXiv2025

Direct and Inverse Problems for Restricted Signed Sumsets -- I

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 $$h_{\pm}A=\left\lbrace \sum_{i=1}^{k} \lambda_{i} a_{i}: \lambda_{i} \in \{-h, \ldots, 0, \ldots, h\} \ \text{for} \ i= 1, 2, \ldots, k \ \text{and} \ \sum_{i=1}^{k} \left|\lambda_{i} \right| =h\right\rbrace,$$ and the restricted $h$-fold signed sumset of $A$, denoted by $h^{\wedge}_{\pm}A$, is defined as $$h^{\wedge}_{\pm}A=\left\lbrace \sum_{i=1}^{k} \lambda_{i} a_{i}: \lambda_{i} \in \left\lbrace -1, 0, 1\right\rbrace \ \text{for} \ i= 1, 2, \ldots, k \ \text{and} \ \sum_{i=1}^{k} \left|\lambda_{i} \right| = h\right\rbrace. $$ 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.

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arXiv2024

Some direct and inverse problems for the Restricted Signed sumset in set of integers

Given a positive integer $h$ and a nonempty finite set of integers $A=\{a_{1},a_{2},\ldots,a_{k}\}$, the restricted $h$-fold signed sumset of $A$, denoted by $h^{\wedge}_{\pm}A$, is defined as $$h^{\wedge}_{\pm}A=\left\lbrace \sum_{i=1}^{k} \lambda_{i} a_{i}: \lambda_{i} \in \left\lbrace -1, 0, 1\right\rbrace \ \text{for} \ i= 1, 2, \ldots, k \ \text{and} \ \sum_{i=1}^{k} \left| \lambda_{i} \right| =h\right\rbrace.$$ The direct problem associated with this sumset is to find the optimal lower bound of $|h^{\wedge}_{\pm}A|$, and the inverse problem associated with this sumset is to determine the structure of the underlying set $A$, when $|h^{\wedge}_{\pm}A|$ attains the optimal lower bound. Bhanja, Komatsu and Pandey studied the direct and inverse problem 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 this paper, we prove these conjectures for $h=4$. We also prove the direct and inverse theorems for arbitrary $h$ under certain restrictions on the set $A$ which are particular cases of the conjectures. Moreover, we prove these conjectures for arithmetic progressions.

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arXiv2017

Restricted-sum-dominant sets

Let $A$ be a nonempty finite subset of an additive abelian group $G$. Define $A + A := \{a + b : a, b \in A\}$ and $A \dotplus A := \{a + b : a, b \in A~\text{and}~ a \neq b\}$. The set $A$ is called a {\em sum-dominant (SD) set} if $|A + A| > |A - A|$, and it is called a {\em restricted sum-domonant (RSD) set} if $|A \dotplus A| > |A - A|$. In this paper, we prove that for infinitely many positive integers $k$, there are infinitely many RSD sets of integers of cardinality $k$. We also provide an explicit construction of infinite sequence of RSD sets.

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