On the generalisation of Roth's theorem

Abstract

We prove a generalisation of Roth's theorem for proper adelic curves, assuming that the logarithmic absolute values of the approximants satisfy a condition similar to the equicontinuity with respect to the places. This work extends Corvaja's results [Cor97] for fields admitting a product formula, and Voita's ones [Voj21] for arithmetic function fields. Adelic curvesWe will use the following notations throughout the whole paper: log + x := max{0, log x} , log − x := min{0, log x} ; ∀x ∈ R>0