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PreprintarXiv licencearXiv

Explicit bounds for the number of solutions to Roth-type inequalities over adelic curves

  • YuXi Li
  • GuanHeng Zhao
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Abstract

In this paper, we provide an upper bound on the number of a special system of Roth-type Diophantine inequalities over proper adelic curves. By refining the technique from previous work, we successfully eliminate the large-height restriction on the solutions. Furthermore, the constraints on the parameters within the inequality are further relaxed. We also give a constructive quantitative generalization of the Vojta-Dolce-Zucconi system, replacing compactness-based extraction by a discrete pigeonhole argument and a finite grid partition, so that the resulting bound has explicit dependence on the relevant parameters.

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Cite

BibTeXRIS
BibTeX
@misc{li2026explicit,
  title = {{Explicit bounds for the number of solutions to Roth-type inequalities over adelic curves}},
  author = {YuXi Li and GuanHeng Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.06973},
  eprint = {2611.06973},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.06973},
}
RIS
TY  - UNPB
TI  - Explicit bounds for the number of solutions to Roth-type inequalities over adelic curves
AU  - YuXi Li
AU  - GuanHeng Zhao
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.06973
UR  - https://arxiv.org/abs/2611.06973
AB  - In this paper, we provide an upper bound on the number of a special system of Roth-type Diophantine inequalities over proper adelic curves. By refining the technique from previous work, we successfully eliminate the large-height restriction on the solutions. Furthermore, the constraints on the parameters within the inequality are further relaxed. We also give a constructive quantitative generalization of the Vojta-Dolce-Zucconi system, replacing compactness-based extraction by a discrete pigeonhole argument and a finite grid partition, so that the resulting bound has explicit dependence on the relevant parameters.
ER  -