Abstract
A function is called elementary if it can be computed in time bounded by a tower of powers of constant height. A trace function for an irrational number \(\alpha \) is a function that maps each rational number r to a rational number that is closer to \(\alpha \) than r is. We show, as was conjectured by Kristiansen, that there exists an irrational number that has an elementary trace function but whose continued fraction expansion is not elementary. We also show that there exists an irrational number that has an elementary trace function but whose sum approximation, i.e., the function that maps each positive integer n to the index of the nth 1 in the binary expansion of the number, is not elementary.
Supported by JSPS KAKENHI Grant Numbers JP18H03203, JP23H03346 and JSPS Bilateral Program Grant Number JPJSBP120204809.
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Hiroshima, K., Kawamura, A. (2023). Elementarily Traceable Irrational Numbers. In: Della Vedova, G., Dundua, B., Lempp, S., Manea, F. (eds) Unity of Logic and Computation. CiE 2023. Lecture Notes in Computer Science, vol 13967. Springer, Cham. https://doi.org/10.1007/978-3-031-36978-0_11
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DOI: https://doi.org/10.1007/978-3-031-36978-0_11
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