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