Simultaneous p-adic Diophantine approximation

Victor Beresnevich, Jason Levesley, Benjamin Ward

Published 2021-01-13Version 1

The goal of this paper is to develop the theory of weighted Diophantine approximation of $p$-adic variables. Firstly, we establish complete analogues of Khintchine's theorem and the Jarn\'ik-Besicovitch theorem for `weighted' simultaneous Diophantine approximation in the $p$-adic case. Secondly, we obtain a lower bound for the Hausdorff dimension of weighted simultaneously approximable points lying on $p$-adic curves. One of the key tools in our proofs is the Mass Transference Principle, including its recent extension due to Wang and Wu. Our result for curves has a natural constraint on the exponents of approximation and, in the case of polynomial curves, complements a theorem of Bugeaud et al, which deals with large exponents.