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On simultaneous uniform approximation to a \(p\)-adic number and its square. (English) Zbl 1204.11105

Let \(p\) be a prime number. The author has shown that a result of O. Teulié, [Acta Arith. 102, No. 2, 137–155 (2002; Zbl 0993.11036)] is nearly best possible by constructing a \(p\)-adic number \(\xi\) such that \(\xi\) and \(\xi^2\) are uniformly simultaneously very well approximable by rational numbers with the same denominator. This result was already obtained by D. Zelo, [”Simultaneous approximation to real and \(p\)-adic numbers,” PhD thesis, Univ. Ottawa (2009), arXiv:0903.0086], but the method of the author by its construction using \(p\)-adic continued fractions is more direct and simpler.
Let \(\lambda = (\sqrt{5}-1)/2\). In the following \(p\) is a prime number and the absolute value \(|.|\) is normalized in such a way that \(|p|_p= p^{-1}\). The Zelo result is:
Let \(\varepsilon\) be a positive real number. There exists a \(p\)-adic number \(\xi\) which is neither rational nor quadratic and a positive real number \(c\) such that the system of inequalities \[ |x_0\xi-x_1|_p\leq cX^{-1-\lambda+\varepsilon}, |x_0\xi^2-\xi_2|_p<cX^{-1-\lambda+\varepsilon}, \max\{|x_0|,|x_1|,|x_2|\}\leq X \] has a non-zero integer solution \((x_0,x_1,x_2)\) for every real number \(X>1\).
Definitions:
1) Let \(a\) and \(b\) be two symbols. Set \(f_1=b\), \(f_2=a\) and let \(f_n=f_{n-1}f_{n-2}\) be the concatenation of the words \(f_{n-1}\) and \(f_{n-2}\) for \(n\geq 3\). Then \(f_\infty\) is the Fibonacci word on the alphabet \(\{a,b\}\).
2) Let \(n\geq 1\) be an integer and let \(\xi\) be a \(p\)-adic number. Let us denote \(\hat{\lambda}_n(\xi)\) the supremum of the real number \(\hat{\lambda}\) such that, for every sufficiently large real number \(X\), the system of inequalities \[ \max_{1\leq m\leq n} |x_0\xi^m-x_m|_p\leq X^{-1-\hat{\lambda}},\;\;0<\max\{|x_0|,|x_1|,\dots,|x_n|\}\leq X \] has a solution in integers \(x_0,\dots,x_n\).
The theorem of Zelo asserts that \[ \sup\{\hat{\lambda}_2(\xi): \xi \in \mathbb Q_p,\;\xi \text{\;is neither rational nor quadratic\;} \}=\lambda. \] The author gives the following constructive proof of this result:
Theorem: Let \(v\) be a positive integer and let \((v_n)_{n\geq 1}\) be the Fibonacci word on \(\{v,v+1\}\) starting with \(v\). Let \(\xi_v\) denote the \(p\)-adic number \[ \xi_v:=1+\lim_{n\rightarrow\infty} \frac{p^{v_1}}{1+\frac{p^{v_2}}{1+\frac{p^{v_3}}{\dots+ p^{v_n}}}} \] Then we have \(\hat{\lambda}_2(\xi_v)\geq (1-7/v)\lambda\) and \(\sup\{\hat{\lambda}_2(\xi_v) \;:\;v\geq 1\}=\lambda\).

MSC:

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

Citations:

Zbl 0993.11036
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References:

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[8] Olivier Teulié, Approximation d’un nombre \?-adique par des nombres algébriques, Acta Arith. 102 (2002), no. 2, 137 – 155 (French). · Zbl 0993.11036
[9] D. Zelo, Simultaneous approximation to real and \( p\)-adic numbers, PhD thesis, Univ. Ottawa, 2009, arXiv:0903.0086.
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