Expander estimates for cubes

If \mathscr{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\mathscr{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} .