Jian Xu 2008 Nonlinearity 21 2113 doi:10.1088/0951-7715/21/9/012
On sums of partial quotients in continued fraction expansions
Jian Xu
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Assume x 
[0, 1) taking on its continued fraction expansion as [a1(x), a2(x), ...]. For any n ≥ 1, write 
. Khintchine (1935 Compos. Math. 1 361–82) proved that Sn(x)/(n log n) converges in measure to 1/log 2 with respect to 
, where denotes the one-dimensional Lebesgue measure. Philipp (1988 Monatsh. Math. 105 195–206) showed that {an(x), n ≥ 1} cannot satisfy a strong law of large numbers for any reasonably growing norming sequence. In (Wu and Xu 2008 Preprint), we discussed the sets of continued fractions whose sums of partial quotients tend to infinity with the polynomial growth rate. In this paper, we consider the sets of continued fractions whose sums of partial quotients tend to infinity exponentially and doubly exponentially. The Hausdorff dimensions of such sets are determined.
- PACS
- MSC
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11A55 Continued fractions (For approximation results, see 11J70) (See also 11K50, 30B70, 40A15)
26A42 Integrals of Riemann, Stieltjes and Lebesgue type (See also 28-XX)
30B70 Continued fractions (See also 11A55, 40A15)
- Subjects
- Dates
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Issue 9 (September 2008)
Received 11 January 2008, in final form 9 July 2008
Published 7 August 2008
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On sums of partial quotients in continued fraction expansions
Jian Xu 2008 Nonlinearity 21 2113
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