On the optimal rate of equidistribution in number fields

25 Oct 2018
•
Fraczyk Mikolaj
•
Szumowicz Anna

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 $\mathcal O_k$.(read more)