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Estimates for partial sums of continued fraction partial quotients. (English) Zbl 0589.10056
Let $$\alpha =[0,a_ 1,a_ 2,...]$$ be the representation of the irrational number $$\alpha$$ as a regular continued fraction and let F be an arithmetic function. Put $$S_ N(F,\alpha):=\sum_{n\leq N}F(a_ n)$$. Khinchin proved that if $$F(r)=O(r^{-\delta})$$, $$r\to \infty$$, for some positive $$\delta$$, then $(1)\quad (1/N) S_ N(F,\alpha)=(1+o(1))\quad (1/\log 2)\sum^{\infty}_{r=1}F(r) \log (1+1/r(r+2)),\quad N\to \infty.$ If F is such that the sum in (1) converges absolutely, then (1) follows from Birkhoff’s individual ergodic theorem. The case $$F(r)=I(r)=r$$ is not covered by these theorems. Khinchin proved that if one truncates the $$a_ n$$ in the following way: $$b_ n=a_ n$$ if $$a_ n<n(\log n)^{4/3}$$ and $$b_ n=0$$ otherwise, then $(2)\quad (1/N \log N)\sum_{n\leq N}b_ n\to 1/\log 2\quad in\quad measure\quad as\quad N\to \infty.$ The limit in (2) cannot hold for almost all $$\alpha$$ since it follows from a theorem of Borel that for almost all $$\alpha$$ one has for infinitely many n that $$n \log n \log \log n<a_ n<n(\log n)^{4/3}.$$
In the present paper it is shown that the obstacle to the almost everywhere convergence of (2) is the occurrence of a single large $$a_ n$$. The main result reads: Let F be a positive valued arithmetic function satisfying $(\sum_{j\leq N}F^ 2(j)/j^ 2)/(\sum_{j\leq N}F(j)/j^ 2)^ 2\quad \leq \quad N(\log N)^{-3/2-\epsilon}$ for some $$\epsilon >0$$. Then for almost all $$\alpha$$ and for $$N>N_ 0(\alpha):$$ $S_ N(F,\alpha)=(1+o(1))(N/\log 2)\sum_{r\leq N}F(r) \log (1+1/r(r+2))+\delta \quad \max_{1\leq n\leq N}F(a_ n),$ with $$\delta =\delta (N,\alpha,F)$$, $$0\leq \delta \leq 1$$. The proof consists of a carefully handling of expressions involving sums of truncated partial quotients. On the basis lies the well-known mixing property of the shift operator T : $$Tx=1/x-[1/x]$$. There are a few further interesting results.
Reviewer: H.Jager

##### MSC:
 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 11A55 Continued fractions
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