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

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.