Sums of dilates of finite sets of integers

Let \(A\) be a subset of integers and let \(2\cdot A+k\cdot A= \{2a_1+ka_2 : a_1,a_2\in A\}\). Y. O. Hamidoune and J. Rué proved that if \(k\) is an odd prime and if \(A\) is a finite set of integers such that \(|A|>8k^k\), then \(|2\cdot A+k\cdot A|\ge (k+2)|A|-k^2-k+2\). I will show how to extend this result to the case when \(k\) is a power of an odd prime.

Zeljka Ljujic