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

Let beta > 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 beta-expansion which contains x is an element of (0,1], denote by I-n(x) and write the length of I-n(x) as vertical bar I-n(x)vertical bar. In this paper, we prove that the extremely irregular set containing points x is an element of [0,1] whose upper limit of -log(beta) vertical bar I-n(x)vertical bar/n equals to 1 + lambda(beta) is residual for every lambda(beta) > 0, where lambda(beta) is a constant depending on beta. (c) 2016 Elsevier Inc. All rights reserved.