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On approximation of \(p\)-adic numbers by \(p\)-adic algebraic numbers. (English) Zbl 1078.11050

Let \(P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0\) be a polynomial in \(\mathbb Z[x]\) with height \(H(P)=\max| a_i| \). Let \(\Psi(x)\) be monotonically decreasing. Let \(M_n(\Psi)\) be the set of \(p\)-adic numbers \(\omega\) such that the inequality \[ | P(\omega)| _p<H(P)^{-n}\Psi(H(P)) \] has infinitely many solutions in polynomials \(P\) of degree at most \(n\). Then \(M_n(\Psi)\) is null or full (with respect to the Haar measure on \(\mathbb Q_p\)) according as \(\sum_{h=1}^\infty \Psi(h)\) converges or diverges. There is a corresponding theorem where \(P(\omega)\) and \(H(P)\) are replaced by \(\omega-\alpha\) and the height of \(\alpha\) for an algebraic number \(\alpha\in\mathbb Q_p\) of degree \(n\). These theorems are \(p\)-adic analogues of the results over \(\mathbb R\) proved by the second author [Acta Arith. 53, No. 1, 17–28 (1989; Zbl 0692.10042)] and the first author [Acta Arith. 90, No. 2, 97–112 (1999; Zbl 0937.11027)].

MSC:

11J83 Metric theory
11J61 Approximation in non-Archimedean valuations
11K41 Continuous, \(p\)-adic and abstract analogues
11K60 Diophantine approximation in probabilistic number theory
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References:

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