54 research outputs found
Upper and lower fast Khintchine spectra in continued fractions
For an irrational number , let be
its continued fraction expansion. Let be a function with as . The
(upper, lower) fast Khintchine spectrum for is defined as the Hausdorff
dimension of the set of numbers for which the (upper, lower) limit
of is equal to . The fast
Khintchine spectrum was determined by Fan, Liao, Wang, and Wu. We calculate the
upper and lower fast Khintchine spectra. These three spectra can be different.Comment: 13 pages. Motivation and details of proofs are adde
Subexponentially increasing sums of partial quotients in continued fraction expansions
We investigate from a multifractal analysis point of view the increasing rate
of the sums of partial quotients , where
is the continued fraction expansion of an
irrational . Precisely, for an increasing function , one is interested in the Hausdorff
dimension of the setsSeveral cases are solved by Iommi and
Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case
. We show that when , has Hausdorff dimension . Thus, surprisingly, the
dimension has a jump from to at . In a
similar way, the distribution of the largest partial quotient is also studied.Comment: 12 pages. More details for the proof of Theorem 1.2. are adde
Multifractal analysis of some multiple ergodic averages for the systems with non-constant Lyapunov exponents
We study certain multiple ergodic averages of an iterated functions system
generated by two contractions on the unit interval. By using the dynamical
coding of the attractor, we compute the Hausdorff
dimension of the set of points with a given frequency of the pattern 11 in
positions .Comment: 13 page
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