Families of Spectral Sets for Bernoulli Convolutions

We study the harmonic analysis of Bernoulli measures μλ, a one-parameter family of compactly supported Borel probability measures on the real line. The parameter λ is a fixed number in the open interval (0,1). The measures μλ may be understood in any one of the following three equivalent ways: as infinite convolution measures of a two-point probability distribution; as the distribution of a random power series; or as an iterated function system (IFS) equilibrium measure determined by the two transformations λ(x±1). For a given λ, we consider the harmonic analysis in the sense of Fourier series in the Hilbert space L2(μλ). For L2(μλ) to have infinite families of orthogonal complex exponential functions e2πis(⋅), it is known that λ must be a rational number of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{m}{2n}$\end{document}, where m is odd. We show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}(\mu_{\frac{1}{2n}})$\end{document} has a variety of Fourier bases; i.e. orthonormal bases of exponential functions. For some other rational values of λ, we exhibit maximal Fourier families that are not orthonormal bases.