Expander estimates for cubes

If $\mathcal{A}$ is a set of natural numbers of exponential density $\delta$, then the exponential density of all numbers of the form $x^3+a$ with $x\in \mathbb{N}$ and $a\in \mathcal{A}$ is at least $\min (1,\frac{1}{3}+\frac{5}{6}\delta )$. This is a considerable improvement on the previous best lower bounds for this problem, obtained by Davenport more than 80 years ago. The result is the best possible for $\delta \ge \frac{4}{5}$.