Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

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.