On the spectra of Cantor measures

TL;DR

This paper characterizes the structure of maximal orthogonal exponential sets in L^2 spaces of Cantor measures with specific contraction factors and digit sets, revealing a tree-based correspondence.

Contribution

It provides a complete characterization of maximal orthogonal exponential sets for certain Cantor measures, linking digit expansions to tree labelings.

Findings

The n+1-th digit in frequency expansions has m possible values.

Maximal orthogonal sets correspond to labelings of an m-homogeneous rooted tree.

The results apply to Cantor measures with contraction factors related to prime powers.

Abstract

We consider Cantor measures on the line, with contraction factor $N^{-1}=p^{-\alpha}$ (where $p$ a positive prime, $\alpha$ a positive integer) and $m$ positive integer digits lying in distinct residue classes modulo $N$. We obtain a complete characterization of maximal orthogonal sets of exponentials in $L^2(\mu)$, for a class of such measures $\mu$. It is proved that the $n+1$-th digit in the base-$N$ expansion of frequencies in a maximal orthogonal set, with the first $n$ digits prescribed, has $m$ possible values. In consequence, there are a correspondence between labelings of the $m$-homogeneous rooted tree and maximal orthogonal sets of frequencies.