Sums of transcendental dilates

Abstract

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

7 pages