Spectral property of Cantor measures with consecutive digits

TL;DR

This paper investigates the spectral properties of Cantor measures with consecutive digits, classifying when they have orthogonal exponential bases, and characterizing maximal orthogonal sets through $q$-adic expansions.

Contribution

It provides a complete classification of spectral Cantor measures with consecutive digits and characterizes all maximal orthogonal sets using $q$-adic tree mappings.

Findings

Measures with $q$ dividing $b$ have a complete orthogonal exponential system.

Maximal orthogonal sets are classified into regular and irregular types.

Examples of orthogonal exponentials with zero Beurling dimension are constructed.

Abstract

We consider equally-weighted Cantor measures $\mu_{q,b}$ 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 $L^2(\mu_{q,b})$. 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 $\Lambda$ when $q$ divides $b$ via a maximal mapping on the $q-$adic tree in which all elements in $\Lambda$ 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 $L^2(\mu_{q,b})$ 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.