Bel, Pierre 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 J. Number Theory 133, No. 10, 3362-3380 (2013). 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. Cited in 3 Documents MSC: 11J13 Simultaneous homogeneous approximation, linear forms 11J61 Approximation in non-Archimedean valuations 11J82 Measures of irrationality and of transcendence Keywords:simultaneous approximation; \(v\)-adic number; number field; algebraic number Citations:Zbl 0186.08603; Zbl 0993.11036; Zbl 1228.11100 PDFBibTeX XMLCite \textit{P. Bel}, J. Number Theory 133, No. 10, 3362--3380 (2013; Zbl 1295.11077) Full Text: DOI References: [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 [4] Laurent, M., Simultaneous rational approximation to the successive powers of a real number, Indag. Math. (N. S.), 14, 45-53 (2003) · Zbl 1049.11069 [5] Roy, D., Approximation to real numbers by cubic algebraic integers I, Proc. London Math. Soc., 88, 42-62 (2004) · Zbl 1035.11028 [6] Roy, D., Approximation to real numbers by cubic algebraic integers II, Annals of Math., 158, 1081-1087 (2003) · Zbl 1044.11061 [7] Roy, D., Approximation simultanée dʼun nombre et de son carré, C. R. Acad. Sci. Paris, 336, 1-6 (2003) · Zbl 1038.11042 [8] Roy, D., On two exponents of approximation related to a real number and its square, Canad. J. Math., 59, 211-224 (2007) · Zbl 1115.11036 [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.