Let [Formula: see text] and the run-length function [Formula: see text] be the maximal length of consecutive zeros amongst the first [Formula: see text] digits in the [Formula: see text]-expansion of [Formula: see text]. The exceptional set [Formula: see text] is investigated, where [Formula: see text] is a monotonically increasing function with [Formula: see text]. We prove that the set [Formula: see text] is either empty or of full Hausdorff dimension and residual in [Formula: see text] according to the increasing rate of [Formula: see text].
For any β > 1, let T β : [0, 1) → [0, 1) be the β-transformation defined by T β x = βx mod1. We study the uniform recurrence properties of the orbit of a point under the β-transformation to the point itself. The size of the set of points with prescribed uniform recurrence rate is obtained. More precisely, for any 0 r +∞, the setif 0 r 1 and is countable if r > 1. In addition, when r = 1, it is uncountable but has zero Hausdorff dimension.
Abstract. 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 − log β |In(x)| n equals to 1 + λ(β) is residual for every λ(β) > 0, where λ(β) is a constant depending on β.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.