Hausdorff dimension of the maximal run-length in dyadic expansion

Zou, Ruibiao

Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 881-888.

Résumé

For any $x\in [0,1)$, let $x=[{ϵ}_1,{ϵ}_2,\cdots ,]$ be its dyadic expansion. Call $r_n(x):=max\{j\ge 1:{ϵ}_{i+1}=\cdots ={ϵ}_{i+j}=1$, $0\le i\le n-j\}$ the $n$-th maximal run-length function of $x$. P. Erdös and A. Rényi showed that $\underset{n\to \mathrm{\infty }}{lim}{r}_n(x)/{\log }_{2}n=1$ almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose run-length function assumes on other possible asymptotic behaviors than ${\log }_{2}n$, is quantified by their Hausdorff dimension.

Détail

DOI : 10.1007/s10587-011-0055-5

Classification : 11K55, 28A78, 28A80
Mots-clés : run-length function; Hausdorff dimension; dyadic expansion