On the saturation number for cubic surfaces

TL;DR

This paper studies the distribution of rational points with few prime factors on specific cubic surfaces using advanced number theory techniques.

Contribution

It applies the weighted sieve, circle method, and universal torsors to analyze prime factor constraints on rational points.

Findings

Established density results for rational points with prime factor restrictions.

Demonstrated the effectiveness of combined sieve and circle method techniques.

Provided new bounds on the number of such points on Fermat and Cayley cubic surfaces.

Abstract

We investigate the density of rational points on the Fermat cubic surface and the Cayley cubic surface whose coordinates have few prime factors. The key tools used are the weighted sieve, the circle method and universal torsors.