Expander estimates for cubes

TL;DR

This paper establishes a new lower bound on the exponential density of numbers of the form x^3 + a, where a belongs to a set with a given exponential density, improving previous results significantly.

Contribution

It provides the first substantial improvement in lower bounds for the exponential density of cubic shifts, extending Davenport's classical work from over 80 years ago.

Findings

New lower bound: at least min(1, 1/3 + 5/6 * δ) for the exponential density.

Improves upon Davenport's 80-year-old bounds.

Result is optimal for δ ≥ 4/5.

Abstract

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 $\min(1, \frac 13+\frac 56 \delta)$. 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 $\delta\ge \frac 45$.