Defining the set of integers in expansions of the real field by a closed discrete set

TL;DR

This paper proves that expanding the real field with a closed discrete set and a function with dense image can define the set of integers, impacting how certain multiplicative subgroups influence definability.

Contribution

It establishes conditions under which expansions of the real field define the set of integers, especially involving closed discrete sets and functions with dense images.

Findings

The structure (b,,+,, ext{f}) defines  for certain functions f.

Expanding the real field with specific multiplicative subgroups can define the integers.

The results connect discrete sets, dense images, and definability of .

Abstract

Let D\subseteq \mathbb{R} be closed and discrete and f:D^n \to \mathbb{R} be such that f(D^n) is somewhere dense. We show that (\mathbb{R},+,\cdot,f) defines the set of integers. As an application, we get that for every a,b \in \mathbb{R} with \log_{a}(b)\notin \mathbb{Q}, the real field expanded by the two cyclic multiplicative subgroups generated by a and b defines the set of integers.