A weak effective Roth's theorem over function fields (Q5928703)

The motivation here is the correspondence between Diophantine approximation and Nevanlinna theory noted by Osgood and by Vojta and the observation that approximation over function fields corresponds to Nevanlinna theory with moving targets. In this context, \textit{J. Wang} [Rocky Mt. J. Math. 26, 1225-1234 (1996; Zbl 0942.11034)] gives a new proof of Roth's Theorem in function fields. However, as remarked in the present note, the dependence of Wang's result on the quantity \((2g-2+2\#S+h(t))\) (thus in particular on the genus of the given curve and on the number of places \(S\) involved) is non-optimal. Here the author efficiently proves an inequality with optimal dependence on this quantity at the cost, however, of weakening the main term so that his result is a function-field generalisation just of Thue's theorem.