Green points in the reals

TL;DR

This paper develops a new model-theoretic expansion of the real numbers incorporating green points, showing it is strongly dependent and that definable open sets are semialgebraic, with applications to logarithmic spirals.

Contribution

It introduces a novel expansion of real closed fields with green points, extending Poizat's theory and analyzing its properties and applications.

Findings

The theory is strongly dependent.

Every definable open set is semialgebraic.

The real field with dense logarithmic spirals satisfies the theory.

Abstract

We construct an expansion of a real closed field by a multiplicative subgroup adapting Poizat's theory of green points. Its theory is strongly dependent, and every open set definable in a model of this theory is semialgebraic. We prove that the real field with a dense family of logarithmic spirals, proposed by Zilber, satisfies our theory.