On the Problems Associated with Sum of Dilates

Let $A, B\subseteq \mathbb{Z}$ be nonempty finite subsets and $k$ be a positive integer. The sum of dilates of $A$ and $B$ is defined as $A+k\cdot B=\{a+kb:a\in A$ and $b\in B\}.$ In case of $A=B,$ Freiman et al. proved that $|A+k\cdot A|\geq 4|A|-4$ for $k\geq 3.$ In this article, we obtain the same bound for the $|p\cdot A+k\cdot A|$ such that $k>p^2,$ ($p$ is prime). We also prove the extended inverse result, for the cardinality of sumset $A+3\cdot A$ under some conditions.