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Gaps, isolated points, and the origin of the ray in the one-dimensional Dirichlet spectrum

  • Maksim Lavrov
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Abstract

We study the one-dimensional Dirichlet spectrum and obtain new results concerning the origin μ of the ray contained in the spectrum and its isolated points. First, we prove the improved upper bound μ≤ (448875+1611√(67773))/46982 = 18.4809…. We also exhibit an explicit gap (L,R) in the Dirichlet spectrum, which yields the improved lower bound μ≥ R = 5.9596…. The proof of the lower bound includes an exact computer-assisted finite search over admissible continued-fraction words. We further prove that all points Zk, k≥ 2, introduced by Ivanov are isolated points of the Dirichlet spectrum. Finally, we determine the asymptotic behavior of the neighboring gaps around Zk and describe the equivalence classes of all irrational numbers whose Dirichlet value is Zk.

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Cite

BibTeXRIS
BibTeX
@misc{lavrov2026gaps,
  title = {{Gaps, isolated points, and the origin of the ray in the one-dimensional Dirichlet spectrum}},
  author = {Maksim Lavrov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03787},
  eprint = {2610.03787},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03787},
}
RIS
TY  - UNPB
TI  - Gaps, isolated points, and the origin of the ray in the one-dimensional Dirichlet spectrum
AU  - Maksim Lavrov
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.03787
UR  - https://arxiv.org/abs/2610.03787
AB  - We study the one-dimensional Dirichlet spectrum and obtain new results concerning the origin μ of the ray contained in the spectrum and its isolated points. First, we prove the improved upper bound μ≤ (448875+1611√(67773))/46982 = 18.4809…. We also exhibit an explicit gap (L,R) in the Dirichlet spectrum, which yields the improved lower bound μ≥ R = 5.9596…. The proof of the lower bound includes an exact computer-assisted finite search over admissible continued-fraction words. We further prove that all points Z_k, k≥ 2, introduced by Ivanov are isolated points of the Dirichlet spectrum. Finally, we determine the asymptotic behavior of the neighboring gaps around Z_k and describe the equivalence classes of all irrational numbers whose Dirichlet value is Z_k.
ER  -