Research article - Combinatorics, Number theory
On direct and inverse problems related to some dilated sumsets
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 99-105.

Let $A$ be a nonempty finite set of integers. For a real number $m$, the set $m·A=\left\{ma:a\in A\right\}$ 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 $|{\lambda }_{1}·A+{\lambda }_{2}·A+\cdots +{\lambda }_{h}·A|$, where ${\lambda }_{1},{\lambda }_{2},\cdots ,{\lambda }_{h}$ 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 $A+k·B=\left\{a+kb:a\in A$ and $b\in B\right\}$. In this article, we obtain the lower bound for the cardinality of $A+k·B$ with $k\ge 3$ and describe sets for which equality holds. We also derive an extended inverse result with some conditions for the sumset $A+3·B$.

Accepted:
Published online:
DOI: 10.5802/crmath.537
Classification: 11B13, 11B75
Keywords: Sum of dilates, direct and inverse problems, additive combinatorics

Ramandeep Kaur 1; Sandeep Singh 1

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