Peck, L. G. Simultaneous rational approximations to algebraic numbers. (English) Zbl 0098.26302 Bull. Am. Math. Soc. 67, 197-201 (1961). Let \(\beta_1, \beta_2, \dots,\beta_n\) be linearly independent numbers of a real algebraic number field of degree \(n+1\). By a study of the units of the field it is shown how to obtain all integral solutions of the set of inequalities: \[ q_0>0, \quad\gcd(q_0,q_1,\dots,q_n)=1,\quad | \beta_j/\beta_0-q_j/q_0| < Cq_0^{-1-1/n} \] for fixed \(C\). In particular, there are no solutions if the fixed number \(C\) is small enough. It is shown, further, that for an appropriate \(C\) there are infinitely many integer solutions of \[ | q_0\beta_j-q_j\beta_0| <C\,q_0^{-1/n}(\log q_0)^{-1/(n-1)}\quad (1\leq j\leq n-1),\quad | q_0\beta_n-q_n\beta_0| <C\,q_0^{-1/n}. \] This last result strengthens a theorem of J. W. S. Cassels and H. P. F. Swinnerton-Dyer [Philos. Trans. R. Soc. Lond., Ser. A 248, 73–96 (1955; Zbl 0065.27905)]. Reviewer: J. W. S. Cassels Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 4 ReviewsCited in 18 Documents MSC: 11J13 Simultaneous homogeneous approximation, linear forms 11J68 Approximation to algebraic numbers Keywords:diophantine approximation Citations:Zbl 0065.27905 PDF BibTeX XML Cite \textit{L. G. Peck}, Bull. Am. Math. Soc. 67, 197--201 (1961; Zbl 0098.26302) Full Text: DOI OpenURL References: [1] J. W. S. Cassels and H. P. F. Swinnerton-Dyer, On the product of three homogeneous linear forms and the indefinite ternary quadratic forms, Philos. Trans. Roy. Soc. London. Ser. A. 248 (1955), 73 – 96. · Zbl 0065.27905 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.