RUN-LENGTH FUNCTION FOR REAL NUMBERS IN β-EXPANSIONS

作者：Zheng, Lixuan; Wu, Min^*

来源：Fractals, 2022, 30(03): 2250033.

DOI：10.1142/S0218348X22500335

• Beta-Expansion
• Run-Length Function
• Hausdorff Dimension

Let beta > 1 and x is an element of (0,1] be two real numbers. For all x is an element of (0,1], the run-length function with respect to x, denoted by r(x)(y, n), is defined as the maximal length of the prefix of the beta-expansion of x amongst the first n digits of the beta-expansion of y. The level set @@@ E-a,E-b = {y is an element of (0, 1] : lim inf(n ->infinity)r(x)(y, n)/log(beta)n = a, lim sup(n ->infinity)r(x)(y, n)/log(beta) n = b} (0 <= a <= b <= + infinity) @@@ is investigated in our paper. We obtain the Hausdorff dimension of E-a,E-b which extends many known results on run-length function in beta-expansions.

• 单位
  广东金融学院