Sums of transcendental dilates

We show that there is an absolute constant $c>0$ such that $\mathrm{\mid }A+\lambda \cdot A\mathrm{\mid }\ge e^{c\sqrt{\log \mathrm{\mid }A\mathrm{\mid }}}\mathrm{\mid }A\mathrm{\mid }$ for any finite subset $A$ of $\mathbb{R}$ and any transcendental number $\lambda \in \mathbb{R}$. By a construction of Konyagin and Laba, this is best possible up to the constant $c$.