Expander estimates for cubes

TL;DR

This work proves sharp expander-type lower bounds for cubes: for an input set $\mathscr A$ with exponential density $\delta$, the set of cubic translates $x^3+a$ with $a\in\mathscr A$ has exponential density at least $\min\left(1, \tfrac{1}{3}+\tfrac{5}{6}\delta\right)$, with equality $\delta_3=1$ for $\delta\ge \tfrac{4}{5}$. The authors combine a refined analysis of quadratic Weyl sums, a two-stage Poisson summation on the major arcs, and delicate control of local factors and cubic Gauss sums, to obtain precise major-arc and minor-arc bounds. They then apply a circle-method/density-augmentation argument to finite sets to translate these analytic bounds into a global expansion result, improving Davenport’s classical bounds and establishing the optimal threshold for $\delta_3$ when $\delta\ge \tfrac{4}{5}$. The techniques extend the Vaughan-Vinogradov circle-method approach to arbitrary input sets, highlighting the role of local factors and Poisson-summed structures in cubic exponential sums. The findings have implications for Waring-type problems and the additive theory of sums of cubes, illustrating how exponential-density inputs drive significant expansion in polynomial shift problems.

Abstract

If $\mathscr A$ is a set of natural numbers of exponential density $δ$, 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 $\min(1, \frac 13+\frac 56 δ)$. 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 $δ\ge \frac 45$.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof