Solutions of x^α+y^α=1 in λ-Cantor sets

Abstract

In this paper, we study the structure of the solutions of x^α+y^α=1 with both coordinates in a λ-Cantor set. We prove that, for any α>1, the intersection with the middle-third Cantor set has positive Hausdorff dimension, strengthening the infinitude question posed by Jiang et al. [J. London Math. Soc. 113 (2026), e70408]. To explore the structure of the solutions, for arbitrary α, we find that their Hausdorff dimension undergoes phase transitions as λ varies: we establish the occurrence of Hausdorff phase transitions by proving the existence of zero-dimensional and positive-dimensional regions.

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BibTeX RIS

BibTeX

@misc{jiang2026solutions,
  title = {{Solutions of $x^α+y^α=1$ in $λ$-Cantor sets}},
  author = {Kan Jiang and Nero Ziyu Li and Lifeng Xi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29793},
  eprint = {2610.29793},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29793},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Solutions of x^α+y^α=1 in λ-Cantor sets
AU  - Kan Jiang
AU  - Nero Ziyu Li
AU  - Lifeng Xi
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29793
UR  - https://arxiv.org/abs/2610.29793
AB  - In this paper, we study the structure of the solutions of x^α+y^α=1 with both coordinates in a λ-Cantor set. We prove that, for any α>1, the intersection with the middle-third Cantor set has positive Hausdorff dimension, strengthening the infinitude question posed by Jiang et al. [J. London Math. Soc. 113 (2026), e70408]. To explore the structure of the solutions, for arbitrary α, we find that their Hausdorff dimension undergoes phase transitions as λ varies: we establish the occurrence of Hausdorff phase transitions by proving the existence of zero-dimensional and positive-dimensional regions.
ER  -

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