On Spectral N-Bernoulli Mmeasure

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

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