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Simultaneous Diophantine approximation in the real, complex and \(p\)-adic fields. (English) Zbl 1221.11159
Simultaneous approximation of a real number \(x\), a complex number \(z\) and a \(p\)-adic number \(w\) is studied. Specifically, the authors study the set of points \((x,z,w) \in {\mathbb R} \times {\mathbb C} \times {\mathbb Q}_p\) for which the system of inequalities \[ | P(x) | \leq H^{-v_1} \Psi^{\lambda_1}(H), \quad | P(z) | \leq H^{-v_2} \Psi^{\lambda_2}(H), \quad | P(w) |_p \leq H^{-v_3} \Psi^{\lambda_3}(H), \] is satisfied for infinitely many integer polynomials \(P\) of degree \(n\). Here, \(H\) denotes the naïve height of the polynomial \(P\), i.e., the maximum absolute value among the coefficients of \(P\). The numbers \(v_i\) are required to be non-negative with \(v_1 + 2 v_2 + v_3 = n-3\) and the \(\lambda_j\)’s are positive with \(\lambda_1 + 2 \lambda + \lambda_3 = 1\). Finally, \(\Psi\) is a monotonically decreasing real function.
The authors show that if \(n \geq 3\) and \(\sum_{H=1}^\infty \Psi(H) < \infty\), then the set of triples \((x,z,w)\) for which the system in question has infinitely many solutions is of measure zero with respect to the product measure of the \(1\)-dimensional Lebesgue measure on \({\mathbb R}\), the \(2\)-dimensional Lebesgue measure on \({\mathbb C}\) and the Haar measure on \({\mathbb Q}_p\). This generalizes a result of F. F. Zheludevich [Acta Arith. 46, 285–296 (1985; Zbl 0545.10040)] by providing an analogue of a classical result of A. Baker [Proc. R. Soc. Lond., Ser. A 292, 92–104 (1966; Zbl 0146.06302)] within the present setup.
From the statement of the result, one would speculate that it should follow from the Borel–Cantelli lemma. This is indeed the case, but in order to apply this lemma, the authors need to split up the problem into nine special cases with even more subcases.

11J83 Metric theory
11J13 Simultaneous homogeneous approximation, linear forms
11J61 Approximation in non-Archimedean valuations
Full Text: DOI
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