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On simultaneous rational approximation to a \(p\)-adic number and its integral powers. (English) Zbl 1264.11058
Let \(p\) denote a prime number and \(|\,\cdot\,|_p\) the usual \(p\)-adic value normalized by \(|\,p\,|_p=p^{-1}\). Let \(n\geq 1\) be an integer and \(\xi\) a \(p\)-adic integer.
Mahler introduced the exponent of Diophantine approximation \(w_n\). Let us denote \(\underline{w_n(\xi)}\) the supremum of the real numbers \(w\) such that, for arbitrary large real numbers \(X\), the inequalities \(0<|x_n\xi^n+\dots+x_1\xi+x_0|_p\leq X^{-w-1}, \;\;\max_{0\leq m\leq n}|x_m|\leq X\) have a solution in integers \(x_0,\dots,x_n\).
Another exponent of Diophantine approximation measuring the quality of the simultaneous rational approximation to a number and its first \(n\) integral powers has been introduced recently in [Y. Bugeaud and M. Laurent, Ann. Inst. Fourier 55, No. 3, 773–804 (2005; Zbl 1155.11333)] for the real case: Let us denote \(\underline{\lambda_n(\xi)}\) the supremum of the real numbers \(\lambda\) such that, for arbitrarily large real numbers \(X\), the inequalities \(0<|x_0|\leq X,\;\max_{1\leq m\leq n} |x_0\xi^m-x_m|_p\leq X^{-\lambda-1}\) have a solution in integers \(x_0,\dots,x_n\). It is known that \(\lambda_n(\xi)\geq 1/n\) from \(p\)-adic version of the Dirichlet theorem, \(\lambda_n(\xi) =\max\{1/n,1/(d-1)\}\) for every positive integer \(n\) and every \(p\)-adic algebraic number \(\xi\) of degree \(d\), and \(\lambda_n(\xi)=1/n\) for almost every \(p\)-adic integer \(\xi\).
The spectrum of a function is the set of values taken by this function on the set of transcendental \(p\)-adic numbers.
The authors address the following question: Let \(n\geq 1\) be an integer. Is the spectrum of the function \(\lambda_n\) equal to \([1/n,\infty]\)?
At first, the authors give the following \(p\)-adic analogue of a result in [Y. Bugeaud, Ann. Inst. Fourier 60, No. 6, 2165–2182 (2010; Zbl 1229.11100)]: let \(n\geq 1\) be an integer. For any real number \(w\geq 2n-1\), there exists uncountably many \(p\)-adic integers \(\xi\) such that \(w_1(\xi)=\dots =w_n(\xi)=w\).
Proceeding as in [N. Budarina, D. Dickinson and J. Levesley, Mathematika 56, No. 1, 77–85 (2010; Zbl 1279.11076)], using K. Mahler’s transference principle [Čas. Mat. Fys. 68, 85–92 (1939; Zbl 0021.10402 and JFM 65.0177.01)], they prove : Let \(n\geq 1\) be an integer and \(\lambda\geq 1\) be a real number. There are uncountably many \(p\)-adic integers \(\xi\), which can be constructed explicitly, such that \(\lambda_n(\xi)=\lambda\). In particular, the spectrum of \(\lambda_n\) contains the interval \([1,\infty]\).
A result in [V. I. Bernik and M. M. Dodson, Metric Diophantine approximation on manifolds. Cambridge: Cambridge University Press (1999; Zbl 0933.11040)] on Hausdorff dimension of the set of \(p\)-adic numbers \(\xi\) with a prescribed value for \(\lambda_n(\xi)\) is for \(n=1\), \(\dim\{\xi\in \mathbb Q_p : \lambda_1(\xi)=\lambda\}=\frac{2}{1+\lambda}\). The authors generalize to: Let \(n\geq 2\) be an integer. Let \(\lambda>n-1\) be a real number. Then \(\dim\{\xi\in \mathbb Q_p : \lambda_n(\xi)=\lambda\}=\frac{2}{n(1+\lambda)}\).

MSC:
11J13 Simultaneous homogeneous approximation, linear forms
11J61 Approximation in non-Archimedean valuations
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