The topological property of the irregular sets on the lengths of basic intervals in beta-expansions

Let β>1 be a real number. A basic interval of order n is a set of real numbers in (0,1] having the same first n digits in their β-expansion which contains x∈(0,1], denote by In(x) and write the length of In(x) as |In(x)|. In this paper, we prove that the extremely irregular set containing points x∈[0,1] whose upper limit of −logβ⁡|In(x)|n equals to 1+λ(β) is residual for every λ(β)>0, where λ(β) is a constant depending on β.