The exceptional sets on the run-length function of beta-expansions

Orateur: Lixuan ZHENG
Type: Séminaire des doctorants
Site: UPEC
Salle: P1 005
Date de début: 08/03/2018 - 11:00
Date de fin: 08/03/2018 - 11:45

Let $\beta > 1$ and the run-length function $r_n(x, \beta)$ be the maximal length of consecutive zeros amongst the first $n$ digits in the $\beta$-expansion of $x \in [0, 1]$. The exceptional set
\[E_\max^\phi = \{ x\in [0,1] : \liminf_{n\rightarrow +\infty} \frac{r_n (x,\beta)}{\phi(n)}=0, \limsup_{n\rightarrow +\infty} \frac{r_n(x,\beta)}{\phi(n)}=+\infty \}\]
is investigated, where $\phi : \mathbb{N}\rightarrow \mathbb{R}^+$ is a monotonically increasing function with $\lim_{n\rightarrow + \infty} \phi (n)=+\infty$. We prove that the set $E^\phi_\max$ is either empty or of full Hausdorff dimension and residual in $[0, 1]$ according to the increasing rate of $\phi$.