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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Académie des sciences
2024-02-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.537/ |
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| author | Kaur, Ramandeep Singh, Sandeep |
| author_facet | Kaur, Ramandeep Singh, Sandeep |
| author_sort | Kaur, Ramandeep |
| collection | DOAJ |
| description | 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 $A+3\cdot B$. |
| format | Article |
| id | doaj-art-8ba02bf2a704495bbcb1dbe587a7e418 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-02-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-8ba02bf2a704495bbcb1dbe587a7e4182025-02-07T11:12:53ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-02-01362G19910510.5802/crmath.53710.5802/crmath.537On direct and inverse problems related to some dilated sumsetsKaur, Ramandeep0Singh, Sandeep1Department of Mathematics, Akal University, Talwandi Sabo - 151302, IndiaDepartment of Mathematics, Akal University, Talwandi Sabo - 151302, IndiaLet $A$ 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 $A+3\cdot B$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.537/Sum of dilatesdirect and inverse problemsadditive combinatorics |
| spellingShingle | Kaur, Ramandeep Singh, Sandeep On direct and inverse problems related to some dilated sumsets Comptes Rendus. Mathématique Sum of dilates direct and inverse problems additive combinatorics |
| title | On direct and inverse problems related to some dilated sumsets |
| title_full | On direct and inverse problems related to some dilated sumsets |
| title_fullStr | On direct and inverse problems related to some dilated sumsets |
| title_full_unstemmed | On direct and inverse problems related to some dilated sumsets |
| title_short | On direct and inverse problems related to some dilated sumsets |
| title_sort | on direct and inverse problems related to some dilated sumsets |
| topic | Sum of dilates direct and inverse problems additive combinatorics |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.537/ |
| work_keys_str_mv | AT kaurramandeep ondirectandinverseproblemsrelatedtosomedilatedsumsets AT singhsandeep ondirectandinverseproblemsrelatedtosomedilatedsumsets |