On spectral N-Bernoulli measures

For 0 < rho < 1 and N > 1 an integer, let mu be the self-similar measure defined by mu(center dot) = Sigma(N-1)(i=0) 1/N mu(rho(-1)(center dot) - i). We prove that L-2(mu) has an exponential orthonormal basis if and only if rho = 1/q for some q > 0 and N divides q. The special case is the Cantor measure with rho = 1/2k and N = 2 [16], which was proved recently to be the only spectral measure among the Bernoulli convolutions with 0 < rho < 1 [4]. (C) 2014 Elsevier Inc. All rights reserved.