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Simultaneous approximation of a \(v\)-adic number and of its square by algebraic numbers. (Approximation simultanée d’un nombre \(v\)-adique et de son carré par des nombres algébriques.) (French. English summary) Zbl 1295.11077

Summary: We generalize to an arbitrary number field the result of H. Davenport and W. M. Schmidt [Acta Arith. 15, 393–416 (1969; Zbl 0186.08603)] about the approximation to a real number and his square by rational numbers with the same denominator, as well as its \(p\)-adic analog due to O. Teulié [Acta Arith. 102, No. 2, 137–155 (2002; Zbl 0993.11036)]. By an extension of D. Roy’s method [Acta Arith. 133, No. 2, 185–197 (2008; Zbl 1228.11100)], we prove that our new result is optimal.

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
11J61 Approximation in non-Archimedean valuations
11J82 Measures of irrationality and of transcendence
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[1] Bombieri, E.; Gubler, W., Height in Diophantine Geometry, New Mathematical Monographs, vol. 4 (2006), Cambridge University Press: Cambridge University Press Cambridge
[2] Bugeaud, Y., On simultaneous uniform approximation to a \(p\)-adic number and its square, Proc. Amer. Math. Soc., 138, 3821-3826 (2010) · Zbl 1204.11105
[3] Davenport, H.; Schmidt, W. M., Approximation to real numbers by algebraic integers, Acta Arith., 15, 393-416 (1969) · Zbl 0186.08603
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[9] Roy, D., On simultaneous rational approximations to a real number, its square, and its cube, Acta Arith., 133, 185-197 (2008) · Zbl 1228.11100
[10] Schmidt, W. M., Diophantine approximations and Diophantine equations, Springer Lecture Notes in Math., vol. 1467 (1991), Springer: Springer Berlin, New York · Zbl 0754.11020
[11] Teulié, O., Approximation dʼun nombre \(p\)-adique par des nombres algébriques, Acta Arith., 102, 137-155 (2002) · Zbl 0993.11036
[12] Zelo, D., Simultaneous approximation to real and \(p\)-adic numbers (2009), Department of Mathematics and Statistic, University of Ottawa, PhD. Thesis
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