arXiv subjects

Mohan

Publications and source records attributed to Mohan.

8 recordsLinked to original sources

arXiv2026

On Small Doubling in Right-Ordered Groups and Baumslag-Solitar Groups-II

Recently, Mohan et al. [Results Math. 80 (2025), No. 4, 122] answered Freiman's $3k-4$ conjecture in right-ordered groups under certain restrictions. In this paper, we take a step further by investigating the structure of nonempty subsets $S$ of a right-ordered group satisfying the small doubling condition $|S^2| = 3|S|-3$. Moreover, we provide a complete characterization of all nonempty finite subsets $S$ of the Baumslag-Solitar group $\mathrm{BS}(1,q)$ (with $q \in \mathbb{Z}$ and $q \neq -1$) for which $|S^2| = 3|S|-3$ and the identity element is the minimum of $S$.

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arXiv2025

Extended inverse results for restricted h-fold sumset in integers

Let $A$ be a finite set of $k$ integers. For $h \leq k$, the restricted $h$-fold sumset $h^{\wedge} A$ is the set of all sums of $h$ distinct elements of $A$. In additive combinatorics, much of the focus has traditionally been on finite integer sets whose sumsets are unusually small (cf.\ Freiman's theorem and its extensions). More recently, Nathanson posed the inverse problem for the restricted sumset $h^{\wedge} A$ when $|h^{\wedge} A|$ is small. For $h \in \{2, 3, 4\}$, this question has already been studied by Mohan and Pandey. In this article, we study the inverse problems for $h^{\wedge} A$ with arbitrary $h \geq 3$ and characterize all possible sets $A$ for certain cardinalities of $h^{\wedge} A$.

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arXiv2024

On additive complements in the complement of a set of natural numbers

Let $A$ be a set of natural numbers. A set $B$, a set of natural numbers, is an additive complement of the set $A$ if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. Erd\H{o}s proposed a conjecture that every infinite set of natural numbers has a sparse additive complement, and in 1954, Lorentz proved this conjecture. This article describes the existence or non-existence of those additive complements of the set $A$ that is a subset of the complement of $A$. We provide a ratio test to verify the existence of such additive complements. In precise, we prove that if $A=\{a_i: i\in \mathbb{N}\}$ is a set of natural numbers such that $a_i 1$, then there exists a set $B\subset \mathbb{N}\setminus A$ such that $B$ is a sparse additive complement of the set $A$.

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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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arXiv2024

On additive complement with special structures

Let $A$ be a set of natural numbers. A set $B$, a set of natural numbers, is said to be an additive complement of the set $A$ if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. This article describes various types of additive complements of the set $A$ such as those additive complement of $A$ that does not intersects $A$, additive complements of the form of the union of disjoint infinite arithmetic progressions, additive complement having various density etc. As an application of this study, we also focus on the structure of sumset of arithmetic progression and geometric progression. Apart from this, for given positive real no. $\alpha \leq 1$ and finite set $A$, we investigate a set $B$ such that it can be written as union of disjoint infinite arithmetic progression and density of $A+B$ is $\alpha$.

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arXiv2024

Freiman's $(3k-4)$-like results for subset and subsequence sums

For a nonempty finite set $A$ of integers, let $S(A) = \left\{ \sum_{b\in B} b: \emptyset \not= B\subseteq A\right\}$ be the set of all nonempty subset sums of $A$. In 1995, Nathanson determined the minimum cardinality of $S(A)$ in terms of $|A|$ and described the structure of $A$ for which $|S(A)|$ is the minimum. He asked to characterize the underlying set $A$ if $|S(A)|$ is a small increment to its minimum size. Problems of such nature are inspired by the well-known Freiman's $3k-4$ theorem. In this paper, some results in the direction of Freiman's $3k-4$ theorem for the set of subset sums $S(A)$ are proved. Such results are also extended to the set of subsequence sums $S(\mathbb{A}) = \left\{ \sum_{b\in \mathbb{B}} b: \emptyset \not= \mathbb{B} \subseteq \mathbb{A} \right\}$ of sequence $\mathbb{A}$, where the notation $\mathbb{B} \subseteq \mathbb{A} $, is used for $\mathbb{B}$ is a subsequence of $\mathbb{A}$. The results are further generalized to a generalization of subset and subsequence sums. The main idea of the proofs of the results is to write the set of subset sums $S(A)$ and the set of subsequence sums $S(\mathbb{A})$ in terms of the $h$-fold sumset $hA$ and the $h$-fold restricted sumset $h^\wedge A$. Such representation also gives other proof of some of the results of Nathanson and Mistri et al.

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arXiv2024

Generalized H-fold sumset and Subsequence sum

Let $A$ and $H$ be nonempty finite sets of integers and positive integers, respectively. The generalized $H$-fold sumset, denoted by $H^{(r)}A$, is the union of the sumsets $h^{(r)}A$ for $h\in H$ where, the sumset $h^{(r)}A$ is the set of all integers that can be represented as a sum of $h$ elements from $A$ with no summand in the representation appearing more than $r$ times. In this paper, we find the optimal lower bound for the cardinality of $H^{(r)}A$, i.e., for $|H^{(r)}A|$ and the structure of the underlying sets $A$ and $H$ when $|H^{(r)}A|$ is equal to the optimal lower bound in the cases $A$ contains only positive integers and $A$ contains only nonnegative integers. This generalizes recent results of Bhanja. Furthermore, with a particular set $H$, since $H^{(r)}A$ generalizes subsequence sum and hence subset sum, we get several results of subsequence sums and subset sums as special cases.

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arXiv2024

Extended inverse theorems for $h$-fold sumsets in integers

Let $h \geq 2$, $k \geq 5$ be integers and $A$ be a nonempty finite set of $k$ integers. Very recently, Tang and Xing studied extended inverse theorems for $hk-h+1 < \left|hA\right| \leq hk+2h-3$. In this paper, we extend the work of Tang and Xing and study all possible inverse theorems for $hk-h+1<\left|hA\right| \leq hk+3h-4$. Furthermore, we give a range of $|hA|$ for which inverse problems are not possible.

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