Multifractal analysis of the growth of sums of digits in the continued fraction expansions

Speaker(s)

Michał Rams

Affiliation

Polska Akademia Nauk

Date

Oct. 18, 2013, 10:15 a.m.

Room

room 5840

Seminar

Seminar of Dynamical Systems Group

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I will present results on the rate of growth of $\sum_{i=1}^n a_i(x)$, where
$\{a_i(x)\}$ is the continued fraction expansion of $x$. A well-known
theorem of Khinchin (1935) states that
\[
\frac {\sum_1^n a_i} {n\log n} \to \frac 1 {\log 2},
\]
where the convergence is in measure. On the other hand, Philipp'88 showed
that there doesn't exist any gauge function $g(n)$ for which the pointwise
convergence
\[
\frac {\sum_1^n a_i(x)} {g(n)} \to 1
\]
would hold Lebesgue-almost everywhere. I will present the more detailed
estimation (in the sense of Hausdorff dimension) for the size of the sets of
points $x$ for which this convergence holds for diferent gauge functions
$g$. The results I'm going to present are in papers [Iommi, Jordan], [Wu,
Xu], [Liao, R], and [Jordan, R].

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