Scaling of spectra of Cantor-type measures and some number theoretic considerations

TL;DR

This paper explores the relationship between number theory and spectral measures of Cantor-type sets, focusing on conditions under which certain natural numbers generate complete Fourier bases for these fractal measures.

Contribution

It introduces new criteria linking number theoretic properties to the spectral completeness of Fourier bases on Cantor-type measures.

Findings

Identifies conditions for odd natural numbers to generate Fourier bases

Establishes connections between number theory and spectral measure properties

Provides insights into harmonic analysis on fractal sets

Abstract

We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number $m$ generates a complete or incomplete Fourier basis for a Cantor-type measure with scale $g$.