On the lengths of basic intervals in beta expansions

Let beta > 1 be a real number and let (epsilon(1)(x, beta), epsilon(2)(x, beta), . . .) be the digit sequence in the beta-expansion of a point x is an element of (0, 1]. This note is concerned with the length of the nth order basic interval containing x, denoted by I-n(x), which consists of those points y is an element of (0, 1] such that epsilon(j)(y, beta) = epsilon(j)(x, beta) for all 1 <= j <= n. We establish a relationship between the length of I-n(x) and the beta-expansion of 1, which enables us to obtain the exact value of the length of I-n(x). As an application, we prove that the growth of the length of I-n(x) is multifractal and that the multifractal spectrum depends on beta.