Roth's theorem on weighted projective lines

Let ${\mathbb P}^1_{\mathfrak q}$ be the weighted projective line with coprime weights ${\mathfrak q}=(q_0,q_1)$ and $L=q_0q_1$. The weighted Veronese map $Φ\colon {\mathbb P}^1_{\mathfrak q}\to {\mathbb P}^1$ satisfies $h\circΦ=L\,{\mathfrak s}$. Pulling back Roth's theorem gives the optimal expone