Expansions of the ordered additive group of real numbers by two discrete subgroups

TL;DR

This paper investigates the logical properties of real number groups expanded by two discrete subgroups, showing decidability for quadratic cases and definability of multiplication for the golden ratio using Ostrowski numeration.

Contribution

It establishes decidability of the theory for quadratic parameters and demonstrates how to define multiplication by the golden ratio within these structures.

Findings

Decidability of the theory when a is quadratic.

Definition of multiplication by the golden ratio in the structure.

Use of Ostrowski numeration system for logical definability.

Abstract

The theory of $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z} a)$ is decidable if $a$ is quadratic. If $a$ is the golden ratio, $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z} a)$ defines multiplication by $a$. The results are established by using the Ostrowski numeration system based on the continued fraction expansion of $a$ to define the above structures in monadic second order logic of one successor. The converse that $(\mathbb{R},<,+,\mathbb{Z},\mathbb{Z} a)$ defines monadic second order logic of one successor, will also be established.