On the lengths of basic intervals in beta expansions

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