Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 40 min ago
PreprintarXiv licencearXiv

Spectral eigenvalue problem of Cantor measures and Artin's primitive root conjecture

  • Xing-Gang He
  • Zhi-Yi Wu
  • Feng-Li Yin
Open at sourcePDFDOICite

Abstract

The eigenvalue problem for a probability measure μ with compact support in R is whether there exist a countable set Λ and a nonzero real t≠ 1 such that both Λ and tΛ are spectra of μ, that is, the family EaΛ={e−2πi aλx:λ∈Λ} is an orthonormal base for L2(μ) for a=1, t. The eigenvalue problem was discovered independently by Strichartz (Str00), Laba and Wang (LW02) for the Cantor measures μ4,{0,1} and μ6,{0,1,2}, respectively. In this paper, we investigate the spectral eigenvalue problem for the general spectral Cantor measure μb,𝒟. This topic is naturally related to elementary number theory. Unexpectedly, however, our main results depend on the theory of integers, especially Artin's primitive root conjecture. To some extent, our results suggest that Artin's primitive root conjecture may hold and confirms some viewpoints implied by Minkowski in (Min57).

Related

  1. Characterization on spectra of self-similar measures and the dual spectral set conjecture
    Li-Xiang An, Yan-Song Fu, Ming-Xuan Jiang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  2. Positive Planar Measures for Spectral Pinching and the Corrected BMV Conjecture
    Trung Hoa Dinh, Cong Trinh Le, Trung Dung Vuong
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  3. A spectral Moore problem and the Yang-Koolen conjecture
    Vishal Gupta
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  4. THE OPTIMAL EIGENVALUE COUNT IN AUBRY'S SPECTRAL SPHERE THEOREM FOR DIMENSIONS AT LEAST FOUR
    Anass Nifa
    HALPreprintCC BY-NC-ND 4.0Full textGeometry & topologyHAL
  5. The principal eigenvalue problem for time-periodic nonlocal equations with drift
    Bertrand Cloez, Adil El Abdouni, Pierre Gabriel
    Journal of Functional AnalysisPeer-reviewedCC BY-NC-ND 4.0Full textAnalysis & PDEsHAL

Cite

BibTeXRIS
BibTeX
@misc{he2026spectral,
  title = {{Spectral eigenvalue problem of Cantor measures and Artin's primitive root conjecture}},
  author = {Xing-Gang He and Zhi-Yi Wu and Feng-Li Yin},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29038},
  eprint = {2609.29038},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29038},
}
RIS
TY  - UNPB
TI  - Spectral eigenvalue problem of Cantor measures and Artin's primitive root conjecture
AU  - Xing-Gang He
AU  - Zhi-Yi Wu
AU  - Feng-Li Yin
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.29038
UR  - https://arxiv.org/abs/2609.29038
AB  - The eigenvalue problem for a probability measure μ with compact support in R is whether there exist a countable set Λ and a nonzero real t≠ 1 such that both Λ and tΛ are spectra of μ, that is, the family E_aΛ={e^−2πi aλx:λ∈Λ} is an orthonormal base for L^2(μ) for a=1, t. The eigenvalue problem was discovered independently by Strichartz (Str00), Laba and Wang (LW02) for the Cantor measures μ_4,{0,1} and μ_6,{0,1,2}, respectively. In this paper, we investigate the spectral eigenvalue problem for the general spectral Cantor measure μ_b,𝒟. This topic is naturally related to elementary number theory. Unexpectedly, however, our main results depend on the theory of integers, especially Artin's primitive root conjecture. To some extent, our results suggest that Artin's primitive root conjecture may hold and confirms some viewpoints implied by Minkowski in (Min57).
ER  -