Roth's theorem on weighted projective lines

Abstract

Let ${\mathbb{P}}_{\mathfrak{q}}^1$ be the weighted projective line with coprime weights $\mathfrak{q}=(q_0,q_1)$ and $L=q_0{q}_1$. The weighted Veronese map $\Phi :{\mathbb{P}}_{\mathfrak{q}}^1\to {\mathbb{P}}^1$ satisfies $h\circ \Phi =L\mathfrak{s}$. 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 $\mathfrak{q}=(2,3)$ 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.