On direct and inverse problems related to some dilated sumsets - Kaur, Ramandeep; Singh, Sandeep (Comptes Rendus. Mathématique)
Let $A$ be a nonempty finite set of integers. For a real number $m$, the set $m· 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 $|λ _1· A+λ _2· A+… +λ _h· A|$, where $λ _1, λ _2, … , λ _h$ 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 $A+k· B=\lbrace a+kb:a\in A$ and $b\in B\rbrace $. In this article, we obtain the lower bound for the cardinality of $A+k· B$ with $k\ge 3$ and describe sets for which equality holds. We also derive an extended inverse result with some conditions for the sumset $A+3· B$.