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Description of Generalized Continued Fractions by Finite Automata

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Number Theory and Related Fields

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 43))

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Abstract

A generalized continued fraction algorithm associates every real number x with a sequence of integers; x is rational iff the sequence is finite. For a fixed algorithm A, call a sequence of integers valid if it is the result of A on some input x 0. We show that, if the algorithm is sufficiently well behaved, then the set of all valid sequences is accepted by a finite automaton.

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Acknowledgements

I am very grateful to the referee for many suggestions that considerably improved the chapter.

Supported by a grant from NSERC.

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Authors and Affiliations

  1. School of Computer Science, University of Waterloo, Waterloo, ON, Canada, N2L 3G1

    Jeffrey Shallit

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  1. Jeffrey Shallit
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Correspondence to Jeffrey Shallit .

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Editors and Affiliations

  1. , School of Math. & Physical Sciences, University of Newcastle, Callaghan, 2308, Australia

    Jonathan M. Borwein

  2. , Department of Computing, Macquarie University, Herring Road, North Ryde, 2109, New South Wales, Australia

    Igor Shparlinski

  3. , School of Math. & Physical Sciences, University of Newcastle, Callaghan, 2308, Australia

    Wadim Zudilin

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In memory of Alf van der Poorten, the sorcerer of continued fractions

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Shallit, J. (2013). Description of Generalized Continued Fractions by Finite Automata. In: Borwein, J., Shparlinski, I., Zudilin, W. (eds) Number Theory and Related Fields. Springer Proceedings in Mathematics & Statistics, vol 43. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6642-0_17

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