Abstract
We have provided a model-theoretic proof for the decidability of the additive structure of integers together with the function f mapping x to \(\lfloor \varphi x\rfloor \) where \(\varphi \) is the golden ratio.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Du, Ch.F., Mousavi, H., Rowland, E., Schaeffer, L., Shallit, J.: Decision algorithms for Fibonacci-automatic words, III: Enumeration and abelian properties. Internat. J. Found. Comput. Sci. 27(8), 943–963 (2016)
Wall, D.D.: Fibonacci Series Modulo \(m\). Am. Math. Monthly 67(6), 525–532 (1960)
Zeckendorf, E.: Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas. Bull. Soc. Roy. Sci. Liège 41, 179–182 (1972)
Conant, G.: There are no intermediate structures between the group of integers and Presburger arithmetic. J. Symb. Log. 83(1), 187–207 (2018)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers., 6th edn. Oxford University Press, Oxford (2008)
Mousavi, H., Rowland, E., Schaeffer, L., Shallit, J.: Decision algorithms for Fibonacci-automatic words, I: Basic results. RAIRO Theor. Inform. Appl. 50(1), 39–66 (2016)
Connell, I.G.: Some properties of Beatty sequences. I. Can. Math. Bull. 2, 190–197 (1959)
Connell, I.G.: Some properties of Beatty sequences. II. Can. Math. Bull. 3, 17–22 (1960)
Kaplan, I., Shelah, S.: Decidability and classification of the theory of integers with primes. J. Symb. Log. 82(3), 1041–1050 (2017)
Büchi, J.R.: “On a decision method in restricted second order arithmetic”, Logic Methodology and Philosophy of Science(Proc. 1960 Internat. Cogr.), 1–11, Stanford Univ. Press, Stanford, Calif., (1962)
Khani, M., Valizadeh, A.N., Zarei, A.: “The additive structure of integers with a floor function”, arXiv:2110.01673
Hieronymi, P.: Expansions of the ordered additive group of real numbers by two discrete subgroups. J. Symb. Log. 81(3), 1007–1027 (2016)
Hieronymi, P.: When is scalar multiplication decidable? Ann. Pure Appl. Logic 170(10), 1162–1175 (2019)
Acknowledgements
We would like to thank Philip Hieronymi, for bringing this question (and some others) to our attention, while we were focused on a different one. The question was initially suggested by him for the second author to work on in an academic visit to Illinois, which finally turned out impossible. We would also like to sincerely thank the anonymous referee whose comments helped us improve this manuscript substantially. His/Her questions and comments lead us to this last version which we regard as much more mature than the earlier ones. We would like to thank Ali Valizadeh whose contribution in [11] lead to enhancing proofs in this version.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Khani, M., Zarei, A. The additive structure of integers with the lower Wythoff sequence. Arch. Math. Logic 62, 225–237 (2023). https://doi.org/10.1007/s00153-022-00846-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-022-00846-2