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On Diophantine approximations of dependent quantities in the $$p$$-adic case. (English. Russian original) Zbl 1037.11047
Math. Notes 73, No. 1, 21-35 (2003); translation from Mat. Zametki 73, No. 1, 22-37 (2003).
Let $$f:\mathbb Z_p\rightarrow\mathbb Z_p$$ be a normal function in the sense of Mahler, and let $$\mu$$ be the Haar measure on $$\mathbb Q_p$$ normalized so that $$\mu(\mathbb Z_p)=p$$. Suppose that $$f''(x)\neq0$$ almost everywhere, and let $$\psi:\mathbb N\rightarrow\mathbb R^+$$ be monotonically decreasing. Let $$\mathcal L_f(\psi)$$ be the set of $$x\in\mathbb Z_p$$ for which $| a_2f(x)+a_1x+a_0|_p<\psi(\max| a_i|)$ has infinitely many solutions $$(a_0,a_1,a_2)\in\mathbb Z^3$$. It is proved that $$\mu(\mathcal L_f(\psi))= 0$$ or $$p$$ according as $$\sum_{h=1}^\infty h^2\psi(h)$$ converges or diverges. The convergence case has been proved earlier (assuming that $$h^2\psi(h)$$ is decreasing) by the second author [Dokl. Nats. Akad. Nauk Belarusi 44, No. 2, 28–30 (2000; Zbl 1177.11067)].

##### MSC:
 11J61 Approximation in non-Archimedean valuations 11J83 Metric theory 11K60 Diophantine approximation in probabilistic number theory 11K41 Continuous, $$p$$-adic and abstract analogues
Zbl 1177.11067
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