Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

TL;DR

This paper investigates the distribution of rational points on higher-dimensional orbifolds with hyperplane divisors, using advanced number theory techniques to address a question about their thin set nature.

Contribution

It introduces a novel approach combining the Hardy-Littlewood circle method with recent breakthroughs in Vinogradov's mean value theorem to analyze rational points on orbifolds.

Findings

Established asymptotic estimates for a mixed powers Waring problem

Connected the distribution of rational points to thin set questions

Applied recent results in Vinogradov's mean value theorem

Abstract

We study the density of rational points on a higher-dimensional orbifold $(\mathbb{P}^{n-1},D)$ when $D$ is a $\mathbb{Q}$-divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy-Littlewood circle method to first study an asymptotic version of Waring's problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov's mean value theorem, due to Bourgain-Demeter-Guth and Wooley.