  We show that there is an absolute constant $c>0$ such that $|A+\lambda\cdot
A|\geq e^{c\sqrt{\log |A|}}|A|$ 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 $c$.
