On Spectral N-Bernoulli Mmeasure

For $0<\rho <1$ and $N>1$ an integer, let $\mu$ be the self-similar measure defined by $\mu (\cdot )={\sum }_{i=0}^{N-1}\frac{1}{N}\mu ({\rho }^{-1}(\cdot )-i)$. We prove that $L^2(\mu )$ has an exponential orthonormal basis if and only if $\rho =\frac{1}{q}$ 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}.