On the optimal rate of equidistribution in number fields

Abstract

Let $k$ be a number field. We study how well can finite sets of ${\mathcal{O}}_k$ equidistribute modulo powers of prime ideals, for all prime ideals at the same time. Our main result states that the optimal rate of equidistribution in ${\mathcal{O}}_k$ predicted by the local contstraints cannot be achieved unless $k=\mathcal{Q}$. We deduce that $\mathcal{Q}$ is the only number field where the ring of integers ${\mathcal{O}}_k$ admits a simultaneous $\mathfrak{p}$-ordering, answering a question of Bhargava. Along the way we establish a non-trivial upper bound on the number of solutions $x\in {\mathcal{O}}_k$ of the inequality $|N_{k/\mathcal{Q}}(x(a-x))|\le X^2$ where $X$ is a positive real parameter and $a\in {\mathcal{O}}_k$ is of norm at least $e^{-B}X$ for a fixed real number $B$. The latter can be translated as an upper bound on the average number of solutions of certain unit equations in ${\mathcal{O}}_k$.

46 pages, minor corrections, proofs of lemma 4.5 and proposition 5.5 rewritten