On simultaneous rational approximation to ap-adic number and its integral powers

Abstract

Abstract. Let p be a prime number. For a positive integer n and a p-adic number ξ, let λ n (ξ) denote the supremum of the real numbers λ such that there are arbitrarily large positive integers q such that ||qξ|| p , ||qξ 2 || p , . . . , ||qξ n || p are all less than q −λ−1 . Here, ||x|| p denotes the infimum of |x − n| p as n runs through the integers. We study the set of values taken by the function λ n .

On simultaneous rational approximation to a $p$-adic number and its integral powers, II

“…The present paper is concerned with the simultaneous approximation to successive integral powers of a given transcendental p-adic number ξ by rational numbers with the same denominator. It is partly motivated by a mistake found in [6]: 1 namely, the definition of the quantity λ n (ξ) given in [6,Definition 1.2] is not accurate and leads to the trivial result that λ n (ξ) is always infinite. Indeed, for a fixed nonzero integer y, the quantity |yξ − x| p can be made arbitrarily small, by taking a suitable integer x.…”

“…. , |x n | ≤ X replace the inequalities 0 < |x 0 | ≤ X occurring in [6,Definition 1.2]. All the results of [6] hold for the exponents λ n as in Definition 1.1, but some of the proofs have to be modified accordingly.…”

“…, |x n | ≤ X replace the inequalities 0 < |x 0 | ≤ X occurring in [6,Definition 1.2]. All the results of [6] hold for the exponents λ n as in Definition 1.1, but some of the proofs have to be modified accordingly. Beside pointing out this mistake, we take the opportunity to considerably extend [6,Theorem 2.3] and [6,Lemma 5.1].…”

“…All the results of [6] hold for the exponents λ n as in Definition 1.1, but some of the proofs have to be modified accordingly. Beside pointing out this mistake, we take the opportunity to considerably extend [6,Theorem 2.3] and [6,Lemma 5.1]. Our main motivation is the determination of the spectrum of λ n , that is, the set of values taken by λ n evaluated at transcendental p-adic numbers.…”

Diophantine inheritance for p-adic measures

“…The subject of p-adic Diophantine approximation started with the work of E. Lutz [24] and has seen numerous advances in different contexts over the years including early work of Mahler [25,26]. We refer the reader to [5] for a comprehensive reference, to [23] and the references therein for work relating to p-adic metric Diophantine approximation on manifolds, and to [4,1,6,8] for recent results.…”