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A tetrachotomy for expansions of the real ordered additive group

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Abstract

Let \(\mathcal {R}\) be an expansion of the ordered real additive group. When \(\mathcal {R}\) is o-minimal, it is known that either \(\mathcal {R}\) defines an ordered field isomorphic to \((\mathbb {R},<,+,\cdot )\) on some open subinterval \(I\subseteq \mathbb {R}\), or \(\mathcal {R}\) is a reduct of an ordered vector space. We say \(\mathcal {R}\) is field-type if it satisfies the former condition. In this paper, we prove a more general result for arbitrary expansions of \((\mathbb {R},<,+)\). In particular, we show that for expansions that do not define dense \(\omega \)-orders (we call these type A expansions), an appropriate version of Zilber’s principle holds. Among other things we conclude that in a type A expansion that is not field-type, every continuous definable function \([0,1]^m \rightarrow \mathbb {R}^n\) is locally affine outside a nowhere dense set.

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Notes

  1. To prove Fact 1.2, the reader can either easily redo the proof of [22, Theorem 1.1] or simply apply [18, Theorem C].

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Acknowledgements

We thank Samantha Xu for useful conversations on the topic of this paper, Kobi Peterzil for answering our questions related to [34], and Chris Miller and Michel Rigo for correspondence around Sect. 8. We thank the anonymous referee for very helpful comments that improved the presentation of this paper.

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  1. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL, 61801, USA

    Philipp Hieronymi

  2. Department of Mathematics, University of California, Irvine, USA

    Erik Walsberg

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Correspondence to Philipp Hieronymi.

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The first author was partially supported by NSF grants DMS-1300402 and DMS-1654725. The second author was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 291111/ MODAG. A preprint of this paper was disseminated under the title “On continuous functions definable in expansions of the ordered real additive group”

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Hieronymi, P., Walsberg, E. A tetrachotomy for expansions of the real ordered additive group. Sel. Math. New Ser. 27, 54 (2021). https://doi.org/10.1007/s00029-021-00668-9

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