On direct and inverse problems related to some dilated sumsets

Ramandeep Kaur; Sandeep Singh

doi:10.5802/crmath.537

Research article - Combinatorics, Number theory

On direct and inverse problems related to some dilated sumsets

Ramandeep Kaur ¹; Sandeep Singh ¹

¹ Department of Mathematics, Akal University, Talwandi Sabo - 151302, India

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 99-105.

Abstract

Let $A$ be a nonempty finite set of integers. For a real number $m$ , the set $m \cdot A = {m a : a \in A}$ denotes the set of $m$ -dilates of $A$ . In 2008, Bukh initiated an interesting problem of finding a lower bound for the sumset of dilated sets, i.e., a lower bound for $| λ_{1} \cdot A + λ_{2} \cdot A + \dots + λ_{h} \cdot A |$ , where $λ_{1}, λ_{2}, \dots, λ_{h}$ are integers and $A$ be a subset of integers. In particular, for nonempty finite subsets $A$ and $B$ , the problem of dilates of $A$ and $B$ is defined as $A + k \cdot B = {a + k b : a \in A$ and $b \in B}$ . In this article, we obtain the lower bound for the cardinality of $A + k \cdot B$ with $k \geq 3$ and describe sets for which equality holds. We also derive an extended inverse result with some conditions for the sumset $A + 3 \cdot B$ .

Received: 2022-09-09
Accepted: 2023-07-06
Published online: 2024-02-02

DOI: 10.5802/crmath.537

Classification: 11B13, 11B75
Keywords: Sum of dilates, direct and inverse problems, additive combinatorics

Author's affiliations:

Ramandeep Kaur ¹; Sandeep Singh ¹

¹ Department of Mathematics, Akal University, Talwandi Sabo - 151302, India

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {On direct and inverse problems related to some dilated sumsets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {99--105},
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     volume = {362},
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     doi = {10.5802/crmath.537},
     language = {en},
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Ramandeep Kaur; Sandeep Singh. On direct and inverse problems related to some dilated sumsets. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 99-105. doi : 10.5802/crmath.537. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.537/

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