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Transactions of the American Mathematical Society

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

 
 

 

A truncated Gauss-Kuz′min 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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Article copyright: © Copyright 1988 American Mathematical Society