Spectral property of Cantor measures with consecutive digits

Abstract

We consider equally-weighted Cantor measures μq,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 L2(μq,b). In particular, if q divides b, the measures have a complete orthogonal exponential system and hence spectral measures. Improving the construction by Dutkay et al. (2009) [3], we 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 L2(μq,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 Dutkay et al. (2011) [4] cannot be removed. For an irregular maximal orthogonal set, we show that under some conditions, its completeness is equivalent to that of the corresponding regularized mapping.