Dimension of a Family of Singular Bernoulli Convolutions

✍ Scribed by K.S. Lau

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 705 KB
• Volume: 116
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1993.1116

✦ Synopsis

Let (\left{X_{n}\right}{n=0}^{\infty}) be a sequence of i.i.d. Bernoulli random variables (i.e., (X{n}) takes values ({0,1}) with probability (\frac{1}{2}) each), let (X=\sum_{n=0}^{\infty} \rho^{n} X_{n}), and let (\mu) be the corresponding probability measure. Erdös-Salem proved that if (\frac{1}{2}<\rho<1), and if (\rho^{-1}) is a P.V. number, then (\mu) is singular. In this paper, we study the algebraic structure of (\rho) and the singularity of the correspondent (\mu) in more detail. We introduce a new class of algebraic numbers containing the P.V. numbers, and make use of the selfsimilar property determined by such numbers to calculate the exact mean-quadraticvariation dimension of (\mu). This dimension is most relevant to Strichartz's recent work on Fourier asymptotics of fractal measures. (\mathbb{C} 1993) Academic Press, Inc.