## 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

Zbl 0993.11036
Full Text:

### References:

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