A result on the maximal length of consecutive 0 digits in beta-expansions

作者：Gao, Xiang; Hu, Hui; Li, Zhihui^*

来源：Turkish Journal of Mathematics, 2018, 42(2): 656-665.

DOI：10.3906/mat-1704-119

• beta-Expansion
• consecutive zero digits
• Hausdorff dimension

Let beta > 1 be a real number. For any x is an element of [0, 1] , let r(n) (x, beta) be the maximal length of consecutive zero digits in the first n digits of the beta-expansion of x. In this note, it is proved that for any 0 < a < b < + infinity, the set
E-a,E-b= {x is an element of [0,1] :lim inf( n ->infinity )r(n)(x, beta)/log(beta)n= a,lim sup( n ->infinity)r(n)(x, beta)/log(beta)n = b}
has the full Hausdorff dimension.

• 出版日期2018
• 单位南昌航空大学; 武汉大学