A truncated Gauss-Kuz′min law

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.