Sharp Szemer\'{e}di-Trotter constructions from arbitrary number fields

Abstract

In this note, we describe an infinite family of sharp Szemer\'{e}di-Trotter constructions. These constructions are cartesian products of arbitrarily high dimensional generalized arithmetic progressions (GAPs), where the bases for these GAPs come from arbitrary number fields over $\mathbb{Q}$. This can be seen as an extension of a recent result of Guth and Silier, who provided similar constructions based on the field $\mathbb{Q}(\sqrt{k})$ for square-free $k$. However, our argument borrows from an idea of Elekes, which produces cartesian products where the parts are of unequal size. This significantly simplifies the analysis and allows us to easily give constructions coming from any number field.