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Green points in the reals

Yilong Zhang

TL;DR

The paper adapts the Hrushovski predimension framework to expand real closed fields by a divisible multiplicative subgroup, yielding a rich theory T^{rich}_B in which all open definable sets are semialgebraic and the real field is the open core. It proves the existence of rich structures via amalgamation, authenticates rotundity as a first-order definable condition, and provides an axiomatic characterization (including EC) that ensures ω-saturation and back-and-forth equivalence. The resulting model (the real field equipped with a dense family of logarithmic spirals) satisfies SC_K and EC, giving a concrete realization of Zilber’s spirals in the real setting and exhibiting strong dependence and near model completeness. The work connects to classical transcendence theory through weak CIT, Mordell–Lang, and Ax–Schanuel contexts, contributing a tamely behaved yet non-d-minimal expansion with robust model-theoretic properties and a precise open-core behavior with potential applications to real analytic geometry and o-minimality interfaces.

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.

Green points in the reals

TL;DR

The paper adapts the Hrushovski predimension framework to expand real closed fields by a divisible multiplicative subgroup, yielding a rich theory T^{rich}_B in which all open definable sets are semialgebraic and the real field is the open core. It proves the existence of rich structures via amalgamation, authenticates rotundity as a first-order definable condition, and provides an axiomatic characterization (including EC) that ensures ω-saturation and back-and-forth equivalence. The resulting model (the real field equipped with a dense family of logarithmic spirals) satisfies SC_K and EC, giving a concrete realization of Zilber’s spirals in the real setting and exhibiting strong dependence and near model completeness. The work connects to classical transcendence theory through weak CIT, Mordell–Lang, and Ax–Schanuel contexts, contributing a tamely behaved yet non-d-minimal expansion with robust model-theoretic properties and a precise open-core behavior with potential applications to real analytic geometry and o-minimality interfaces.

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.
Paper Structure (30 sections, 45 theorems, 99 equations)

This paper contains 30 sections, 45 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hrushovski construction
  4. Predimension
  5. Amalgamation
  6. Facts in algebra
  7. Construction
  8. Setup
  9. Existence of a rich structure
  10. Localization
  11. Rotundity
  12. Variety and transcendence
  13. Definability of rotundity
  14. Generic hyperplanes
  15. Rich structures
  16. ...and 15 more sections

Key Result

Theorem A

If SC$_K$ holds for $K=\mathbb{Q}(\beta i)$, then $(\overline{\mathbb{R}},G)$ is a model of $T^{\mathop{\mathrm{rich}}\nolimits}_\mathcal{B}$.

Theorems & Definitions (101)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.7
  • Definition 2.9
  • Definition 2.10
  • ...and 91 more