Transcendental \(p\)-adic continued fractions. (English) Zbl 1388.11040
This paper considers the arithmetic nature of Ruban \(p\)-adic continued fractions [A. A. Ruban, Sib. Mat. Zh. 11, 222–227 (1970; Zbl 0188.10704)]. By work of L. Wang [Sci. Sin., Ser. A 28, 1009–1017, 1018–1023 (1985; Zbl 0628.10036)] and V. Laohakosol [J. Aust. Math. Soc., Ser. A 39, 300–305 (1985; Zbl 0582.10021)] it is known that rational numbers have finite or ultimate periodic \(p\)-adic continued fraction expansions. Here, the author proves that an analogue to Lagrange’s theorem is not true in general, i.e., that quadratic irrationalities have ultimately periodic expansions only under additional hypotheses. Moreover, he shows that that a non-ultimately periodic \(p\)-adic continued fraction has either a transcendental or a quadratic irrational value if it has finite periodic strings of rapidly increasing length. It seems to be difficult to distinguish these two possibilities; for two very special cases, transcendence is proven.
Besides of classical formulae for continued fractions, the main tool of proof is a \(p\)-adic version of Roth’s theorem [M. Hindry and J. H. Silverman, Diophantine geometry. An introduction. New York, NY: Springer (2000; Zbl 0948.11023)].
Besides of classical formulae for continued fractions, the main tool of proof is a \(p\)-adic version of Roth’s theorem [M. Hindry and J. H. Silverman, Diophantine geometry. An introduction. New York, NY: Springer (2000; Zbl 0948.11023)].
Reviewer: Jürgen Wolfart (Frankfurt am Main)
MSC:
| 11J81 | Transcendence (general theory) |
| 11J70 | Continued fractions and generalizations |
| 11J61 | Approximation in non-Archimedean valuations |
References:
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