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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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##### References:
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