Roth's theorem on weighted projective lines
Abstract
Let ℙ1q be the weighted projective line with coprime weights q=(q0,q1) and L=q0q1. The weighted Veronese map Φ: ℙ1q→ ℙ1 satisfies h∘Φ=L s. Pulling back Roth's theorem gives the optimal exponent 2L for the full rational orbit set. For points represented by coprime integral coordinates, counting suggests the smaller exponent q0+q1. We formulate this as a conjecture and prove an S-adic radical form under the abc conjecture. For q=(2,3) the resulting inequality at the target [1:1] is the coprime Hall-type inequality, and Danilov--Elkies examples show that the exponent $5$ is sharp. We also prove the expected exponent unconditionally over characteristic-zero function fields for constant targets and polynomial approximants, and for almost every real target. A weighted abc formulation is introduced; in signature (3,2) finitely many coefficient pairs are equivalent to abc, and weighted Belyi maps transfer this input to arbitrary algebraic targets. Applications include a pullback form of Schmidt's subspace theorem, weighted Thue inequalities, and a counterexample to a naive weighted rescaling of Roth's index construction.
Related
- A 3× 3 counterexample to Lin and Wimmer's rank-minimization conjecture associated with Roth's similarity theorem
- On the Quantitative Bounds of Nonlinear Roth Theorem
- Nef Hattori classes and products of projective lines
- Polarized cylinders on blow-ups of weighted projective planes
- Rigidity of projective area widths and a weighted Crofton formula on RP3
BibTeX
@misc{salami2026roths,
title = {{Roth's theorem on weighted projective lines}},
author = {Sajad Salami},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17589},
eprint = {2610.17589},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17589},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Roth's theorem on weighted projective lines
AU - Sajad Salami
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17589
UR - https://arxiv.org/abs/2610.17589
AB - Let ℙ^1_q be the weighted projective line with coprime weights q=(q_0,q_1) and L=q_0q_1. The weighted Veronese map Φ: ℙ^1_q→ ℙ^1 satisfies h∘Φ=L s. Pulling back Roth's theorem gives the optimal exponent 2L for the full rational orbit set. For points represented by coprime integral coordinates, counting suggests the smaller exponent q_0+q_1. We formulate this as a conjecture and prove an S-adic radical form under the abc conjecture. For q=(2,3) the resulting inequality at the target [1:1] is the coprime Hall-type inequality, and Danilov--Elkies examples show that the exponent $5$ is sharp. We also prove the expected exponent unconditionally over characteristic-zero function fields for constant targets and polynomial approximants, and for almost every real target. A weighted abc formulation is introduced; in signature (3,2) finitely many coefficient pairs are equivalent to abc, and weighted Belyi maps transfer this input to arbitrary algebraic targets. Applications include a pullback form of Schmidt's subspace theorem, weighted Thue inequalities, and a counterexample to a naive weighted rescaling of Roth's index construction.
ER -