The Hilbert Property for integral points of affine smooth cubic surfaces

TL;DR

This paper proves that the set of $S$-integral points on smooth cubic surfaces over number fields is not thin, showing stronger density results and linking to classical problems like the Fermat cubic surface.

Contribution

It establishes the non-thinness of $S$-integral points on smooth cubic surfaces and improves existing density results, connecting to classical number theory problems.

Findings

The set of $S$-integral points is not thin for suitable $k$ and $S$.

The $S$-integral points are Zariski dense, with stronger formality.

Rational integer points on the Fermat cubic surface form a non-thin set.

Abstract

In this paper we prove that the set of $S$-integral points of the smooth cubic surfaces in $\mathbb{A}^3$ over a number field $k$ is not thin, for suitable $k$ and $S$. As a corollary, we obtain results on the complement in $\mathbb{P}^2$ of a smooth cubic curve, improving on Beukers' proof that the $S$-integral points are Zariski dense, for suitable $S$ and $k$. With our method we reprove Zariski density, but our result is more powerful since it is a stronger form of Zariski density. We moreover prove that the rational integer points on the Fermat cubic surface $x^3+y^3+z^3=1$ form a non-thin set and we link our methods to previous results of Lehmer, Miller-Woollett and Mordell.