Ill-distributed sets over global fields and exceptional sets in diophantine geometry
Volume 199 / 2021
Abstract
Let $K\subseteq \mathbb {R}$ be a number field. Using techniques of discrete analysis, we prove that for definable sets $X$ in $\mathbb {R}_{\exp }$ of dimension at most $2$ a conjecture of Wilkie about the density of rational points is equivalent to the fact that $X$ is badly distributed at the level of residue classes for many primes of $K$. This provides a new strategy to prove this conjecture of Wilkie. In order to prove our result, we are led to study an inverse problem as in the works of Walsh (2012, 2014), but in the context of number fields, or more generally global fields. Specifically, we prove that if $K$ is a global field, then every subset $S\subseteq \mathbb {P}^{n}(K)$ consisting of rational points of projective height bounded by $N$, occupying few residue classes modulo $\mathfrak {p}$ for many primes $\mathfrak {p}$ of $K$, must essentially lie in the solution set of a polynomial equation of degree $\lesssim (\log (N))^{C}$ for some constant $C$.
