Spectra of Symmetric Self-similar Measures as Multipliers in Lp

Let mu(theta, D) be the self-similar measure on R satisfying that mu(theta, D) := 1/#D Sigma(j is an element of D) mu(theta, D) circle phi(-1)(j), where phi(j)(x) = theta(-1) x + j, theta > 1, j is an element of D subset of Z, D = -D, and #D denotes the cardinality of the set D. In this work, we will show that, under a mild condition, the closure {(mu(theta, D)) over cap (r n) : n is an element of Z} of the set of Fourier transforms ((mu(theta, D)) over cap (r n) : n is an element of Z} of the self-similar measure mu(theta, D) parameterized by a Pisot number theta is countable for all positive r is an element of Q(theta) but uncountable for Lebesgue-a.e. r > 0. As an application, this, together with results of Sarnak [19] and Hu [8], proves that, for every fixed theta > 1 and the digit set D which is either D = +/-{0, 1, ..., q} or D = +/-{1, 3, ..., 2q - 1}where q is an element of N, the spectrum of the convolution operator f bar right arrow mu(theta, D) * f in L-p (T) (where T is the circle group) is countable and is the same for all p is an element of (1, infinity), that is, {(mu(theta, D)) over cap (n) : n is an element of Z}. This extends the corresponding results of Erdos [6], Salem [17], and Sidorov and Solomyak [21] for Bernoulli convolutions.