On direct and inverse problems related to some dilated sumsets
Let $A$ be a nonempty finite set of integers. For a real number $m$, the set $m\cdot A=\lbrace ma: a\in A\rbrace $ 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\cdot...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2024-02-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.537/ |
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| Summary: | Let $A$ be a nonempty finite set of integers. For a real number $m$, the set $m\cdot A=\lbrace ma: a\in A\rbrace $ 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 $|\lambda _1\cdot A+\lambda _2\cdot A+\cdots +\lambda _h\cdot A|$, where $\lambda _1, \lambda _2, \dots , \lambda _h$ 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 $A+k\cdot B=\lbrace a+kb:a\in A$ and $b\in B\rbrace $. In this article, we obtain the lower bound for the cardinality of $A+k\cdot B$ with $k\ge 3$ and describe sets for which equality holds. We also derive an extended inverse result with some conditions for the sumset $A+3\cdot B$. |
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| ISSN: | 1778-3569 |