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arXiv · 2409.16795

Expander estimates for cubes

Joerg Bruedern·Simon L Rydin Myerson
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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 $δ\ge \frac 45$.

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Joerg Bruedern, Simon L Rydin Myerson. 2025-12-30. Expander estimates for cubes. https://arxiv.org/abs/2409.16795

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