Sumset problem on dilated sets of integers

Abstract

Let A be a non-empty finite set of integers. For integers m and k, let $m\cdot A+k\cdot A=\{m{a}_1+k{a}_2:a_1,a_2\in A\}$. For $m=2$ and an odd prime k such that $\left|A\right|>8k^k$, Hamidoune et al. [6] proved that $|2\cdot A+k\cdot A|\ge (k+2)\left|A\right|-k^2-k+2$. Ljujic [7] extended this result and obtained the same bound for k to be a power of an odd prime and product of two distinct odd primes. Balog et al. [1] proved that $|p\cdot A+q\cdot A|\ge (p+q)\left|A\right|-{\left(pq\right)}^{(p+q-3)(p+q)+1}$, where $p<q$ are relatively primes. In this article, for any odd values of k and under some certain conditions on set A, we obtain that $|2\cdot A+k\cdot A|\ge (k+2)\left|A\right|-2k\left|\hat{A}\right|$, where $\hat{A}$ is the projection of A in $Z/kZ$. This obtained bound is better than the bound given by Balog et al. We also generalize this bound for $|p\cdot A+k\cdot A|$, where p is any odd prime and k be an odd positive integer with $(p,k)=1$.