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

Philipp Hieronymi

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arXiv · 0906.4972

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

Philipp Hieronymi

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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.

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BibTeX RIS

Philipp Hieronymi. 2009-06-26. Defining the set of integers in expansions of the real field by a closed discrete set. https://arxiv.org/abs/0906.4972

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