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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A truncated Gauss-Kuzmin law


Author: Doug Hensley
Journal: Trans. Amer. Math. Soc. 306 (1988), 307-327
MSC: Primary 11K36; Secondary 11A55, 11H41
DOI: https://doi.org/10.1090/S0002-9947-1988-0927693-3
MathSciNet review: 927693
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Abstract: The transformations $ {T_n}$ which map $ x \in [0,\,1)$ onto 0 (if $ x \leqslant 1/(n + 1)$), and to $ \{ 1/x\} $ otherwise, are truncated versions of the continued fraction transformation $ T:x \to \{ 1/x\} $ (but $ 0 \to 0$).

An analog to the Gauss-Kuzmin result is obtained for these $ {T_n}$, and is used to show that the Lebesgue measure of $ T_n^{ - k}\{ 0\} $ approaches $ 1$ exponentially. From this fact is obtained a new proof that the ratios $ \nu /k$, where $ \nu $ denotes any solution of $ {\nu ^2} \equiv - 1\bmod k$, are uniformly distributed $ \bmod 1$ in the sense of Weyl.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0927693-3
Article copyright: © Copyright 1988 American Mathematical Society