On Spectral N-Bernoulli Mmeasure

For $0<\rho<1$ and $N>1$ an integer, let $\mu$ be the self-similar measure defined by $\mu(\cdot)=\sum_{i=0}^{N-1}\frac 1N\mu(\rho^{-1}(\cdot)-i)$. We prove that $L^2(\mu)$ has an exponential orthonormal basis if and only if $\rho=\frac 1q$ for some $q>0$ and $N$ divides $q$. The special case is the Cantor measure with $\rho =\frac 1{2k}$ and $N=2$ \cite {JP}, which was proved recently to be the only spectral measure among the Bernoulli convolutions with $0<\rho<1$ \cite {D}.