On upper and lower fast Khintchine spectra in continued fractions

Abstract: Let $\psi:\mathbb{N}\to \mathbb{R}^+$ be a function satisfying $\psi(n)/n\to \infty$ as $n \to \infty$. We investigate from a multifractal analysis point of view the growth speed of the sums $\sum^n_{k=1}\log a_k(x)$ with respect to $\psi(n)$, where $x=[a_1(x),a_2(x),\cdots]$ denotes the continued fraction expansion of $x\in (0,1)$. The (upper, lower) fast Khintchine spectrum is defined as the Hausdorff dimension of the set of points $x\in(0,1)$ for which the (upper, lower) limit of $\frac{1}{\psi(n)}\sum^n_{k=1}\log a_k(x)$ is equal to $1$. These three spectra have been studied by Fan, Liao ,Wang \& Wu (2013, 2016), Liao \& Rams (2016). In this talk, we will give a new look at the fast Khintchine spectrum, and provide a full description of upper and lower fast Khintchine spectra. The latter improves a result of Liao and Rams (2016).

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

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