Simultaneous p-adic Diophantine approximation

The aim of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to p-adic numbers. Firstly, we establish complete analogues of Khintchine’s theorem, the Duffin–Schaeffer theorem and the Jarník–Besicovitch theorem for ‘weighted’ simultaneous Diophantine approx...

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Bibliographic Details
Published in:Mathematical proceedings of the Cambridge Philosophical Society 2023-07, Vol.175 (1), p.13-50
Main Authors: BERESNEVICH, V., LEVESLEY, J., WARD, B.
Format: Article
Language:English
Subjects:
Approximation
Lower bounds
Mathematical analysis
Theorems
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Summary:The aim of this paper is to develop the theory of weighted Diophantine approximation of rational numbers to p-adic numbers. Firstly, we establish complete analogues of Khintchine’s theorem, the Duffin–Schaeffer theorem and the Jarník–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 manifolds. This is valid for very general classes of curves and manifolds and have natural constraints on the exponents of approximation. The key tools we use in our proofs are the Mass Transference Principle, including its recent extension due to Wang and Wu in 2019, and a Zero-One law for weighted p-adic approximations established in this paper.
ISSN:0305-0041
1469-8064
DOI:10.1017/S0305004122000470