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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A truncated Gauss-Kuz′min law
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by Doug Hensley PDF
Trans. Amer. Math. Soc. 306 (1988), 307-327 Request permission

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.
References
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Additional Information
  • © Copyright 1988 American Mathematical Society
  • 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