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Proceedings of the American Mathematical Society Series B

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When is an automatic set an additive basis?


Authors: Jason Bell, Kathryn Hare and Jeffrey Shallit
Journal: Proc. Amer. Math. Soc. Ser. B 5 (2018), 50-63
MSC (2010): Primary 11B13; Secondary 11B85, 68Q45, 28A80
DOI: https://doi.org/10.1090/bproc/37
Published electronically: August 2, 2018
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Abstract: We characterize those $ k$-automatic sets $ S$ of natural numbers that form an additive basis for the natural numbers, and we show that this characterization is effective. In addition, we give an algorithm to determine the smallest $ j$ such that $ S$ forms an additive basis of order $ j$, if it exists.


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Additional Information

Jason Bell
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: jpbell@uwaterloo.ca

Kathryn Hare
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: kehare@uwaterloo.ca

Jeffrey Shallit
Affiliation: School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Email: shallit@uwaterloo.ca

DOI: https://doi.org/10.1090/bproc/37
Keywords: Additive basis, automatic set, finite-state automaton, Cantor sets
Received by editor(s): October 23, 2017
Received by editor(s) in revised form: March 27, 2018
Published electronically: August 2, 2018
Additional Notes: Research of the first author was supported by NSERC Grant 2016-03632.
Research of the second author was supported by NSERC Grant 2016-03719.
Rsearch of the third author was supported by NSERC Grant 105829/2013.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2018 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)

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