Continuum many spectra for singular continuous spectral measures

Prove that every singular continuous spectral measure has a continuum of spectra, up to translations.

Background

The paper notes that several known singular continuous spectral measures have spectra whose cardinality is continuum, in sharp contrast with the more rigid behavior of Lebesgue spectral measures. It identifies the extension of this phenomenon to all singular continuous spectral measures as a conjectural direction.

This issue is separate from the spectral eigenvalue problem: it concerns the number and structure of spectra available for a fixed measure rather than the scalings of one fixed spectrum.

References

It is conjectured that this is true for any singular continuous spectral measure.

— Spectral eigenvalue problem of Cantor measures and Artin's primitive root conjecture  (2609.29038 - He et al., 24 Sep 2026) in Section 1 (Introduction), paragraph following the discussion of spectral measures