Multifractal analysis of the convergence exponent in continued fractions

Abstract

Let $x\in [0,1)$ be a real number and denote its continued fraction expansion by $[a_1(x),a_2(x),a_3(x),\cdots ]$. The convergence exponent of these partial quotients is defined as \[ τ(x):= \inf\left\{s \geq 0: \sum_{n \geq 1} a^{-s}_n(x)<\infty\right\}. \] In this paper, we investigate some fundamental properties and multifractal analysis of the exponent $\tau (x)$.

17 pages, 1 figure

References in corpus (2)

• Fractal analysis for sets of non-differentiability of Minkowski's question mark function
• On the fast Khintchine spectrum in continued fractions

Cited by in corpus (1)

• Some exceptional sets of Borel-Bernstein Theorem in continued fractions