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Spectral property of Cantor measures with consecutive digits

We consider equally-weighted Cantor measures $μ_{q,b}$ arising from iterated function systems of the form ${b^{-1}(x+i)}$, $i=0,1,...,q-1$, where $q<b$. We classify the $(q,b)$ so that they have infinitely many mutually orthogonal exponentials in $L^2(μ_{q,b})$. In particular, if $q$ divides $b$, the measures have a complete orthogonal exponential system and hence spectral measures. We then characterize all the maximal orthogonal sets $Λ$ when $q$ divides $b$ via a maximal mapping on the $q-$adic tree in which all elements in $Λ$ are represented uniquely in finite $b-$adic expansions and we can separate the maximal orthogonal sets into two types: regular and irregular sets. For a regular maximal orthogonal set, we show that its completeness in $L^2(μ_{q,b})$ is crucially determined by the certain growth rate of non-zero digits in the tail of the $b-$adic expansions of the elements. Furthermore, we exhibit complete orthogonal exponentials with zero Beurling dimensions. These examples show that the technical condition in Theorem 3.5 of \cite{[DHSW]} cannot be removed. For an irregular maximal orthogonal set, we show that under some condition, its completeness is equivalent to that of the corresponding regularized mapping.