Estimates for partial sums of continued fraction partial quotients.

*(English)*Zbl 0589.10056Let \(\alpha =[0,a_ 1,a_ 2,...]\) 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.

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 |