On the optimal rate of equidistribution in number fields

TL;DR

This paper investigates the limits of equidistribution of algebraic integers in number fields, proving that only the rational numbers achieve the optimal rate, and establishes bounds on solutions to certain norm inequalities.

Contribution

It demonstrates that the optimal equidistribution rate is only achievable in the rational numbers, answering Bhargava's question, and provides bounds on solutions to norm inequalities in number fields.

Findings

Only the rational numbers admit a simultaneous p-ordering in their ring of integers.

Established an upper bound on solutions to norm inequalities involving algebraic integers.

Connected solutions of norm inequalities to bounds on solutions of unit equations.

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 $\frak p$-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 $|N_{k/\mathcal Q}(x(a-x))|\leq X^2$ 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 $\mathcal O_k$.