Infinitely many spectral and non-spectral eigenvalues

Establish that every singular spectral pair in the real line has infinitely many spectral eigenvalues and infinitely many non-spectral eigenvalues.

Background

The paper studies the spectral eigenvalue problem: given a singular spectral pair consisting of a measure and one of its spectra, determine which nonzero real scalings of the spectrum remain spectra. Conjecture 1 proposes a general dichotomy asserting the existence of infinitely many scalings of both types.

The later results establish infinitely many primitively non-spectral integer eigenvalues for canonical spectral pairs, but the full conjecture is not proved in the generality stated.

References

Let $(\mu, )$ be a singular spectral pair in $$. Then there are infinitely many spectral eigenvalues and infinitely many non spectral eigenvalues.

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

There are infinitely many primitively spectral integer eigenvalues of the canonical spectral pair $(\mu_{b, }, (b, ))$.

— Spectral eigenvalue problem of Cantor measures and Artin's primitive root conjecture  (2609.29038 - He et al., 24 Sep 2026) in Conjecture 2, immediately after the theorem proving infinitely many primitively non-spectral eigenvalues