Summary: Let \(\{X_ n\}^ \infty_{n=0}\) be a sequence of i.i.d. Bernoulli random variables (i.e. \(X_ n\) takes values \(\{0,1\}\) with probability \(1\over 2\) each), let \(X = \sum^ \infty_{n=0} \rho^ n X_ n\), and let \(\mu\) be the corresponding probability measure. Erdős-Salem proved that if \({1\over 2} < \rho < 1\), and if \(\rho^{-1}\) is a P.V. number, then \(\mu\) is singular. 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 self-similar property determined by such numbers to calculate the exact mean-quadratic-variation dimension of \(\mu\). This dimension is most relevant to Strichartz’s recent work on Fourier asymptotics of fractal measures.