On the Problems Associated with Sum of Dilates

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