{
"id": "2101.05251",
"version": "v1",
"published": "2021-01-13T18:32:31.000Z",
"updated": "2021-01-13T18:32:31.000Z",
"title": "Simultaneous p-adic Diophantine approximation",
"authors": [
"Victor Beresnevich",
"Jason Levesley",
"Benjamin Ward"
],
"comment": "26 pages",
"categories": [
"math.NT"
],
"abstract": "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.",
"revisions": [
{
"version": "v1",
"updated": "2021-01-13T18:32:31.000Z"
}
],
"analyses": {
"keywords": [
"simultaneous p-adic diophantine approximation",
"simultaneously approximable points lying",
"mass transference principle",
"establish complete analogues",
"adic curves"
],
"note": {
"typesetting": "TeX",
"pages": 26,
"language": "en",
"license": "arXiv",
"status": "editable"
}
}
}