Multi-fold sums from a set with few products

In this paper we show that for any $k\ge 2$, there exist two universal constants $C_k,D_k>0$, such that for any finite subset $A$ of positive real numbers with $\mathrm{\mid }AA\mathrm{\mid }\le M\mathrm{\mid }A\mathrm{\mid }$, $|kA|\geq \frac{C_k}{M^{D_k}}\cdot|A|^{\log_42k}.$