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Transactions of the American Mathematical Society

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Diophantine approximation properties of certain infinite sets


Author: Wolfgang M. Schmidt
Journal: Trans. Amer. Math. Soc. 278 (1983), 635-645
MSC: Primary 10F10; Secondary 10K15
DOI: https://doi.org/10.1090/S0002-9947-1983-0701515-3
MathSciNet review: 701515
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Abstract: We exhibit various infinite sets of reals whose finite subsets do not have good simultaneous rational approximations. In particular there is an infinite set such that each finite subset is "badly approximable" in the sense that Dirichlet's theorem is best possible up to a multiplicative constant.


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  • [1] J. W. S. Cassels, On a result of Marshall Hall, Mathematika 3 (1956), 109-110. MR 0084526 (18:875d)
  • [2] -, An introduction to diophantine approximation, Cambridge Tracts 45, Cambridge Univ. Press, 1957. MR 0087708 (19:396h)
  • [3] H. Davenport, A note on diophantine approximation, Studies in Math. Analysis and Related Topics, Stanford Univ. Press, Stanford, Calif., 1962, pp. 77-81. MR 0146145 (26:3671)
  • [4] A. Khintchine, Über eine Klasse linearer Diophantischer Approximationen, Rend. Circ. Mat. Palermo 50 (1926), 170-195.
  • [5] -, Zur metrischen Theorie der diophantischen Approximationen, Math. Z. 24 (1926), 706-714. MR 1544787
  • [6] J. Mycielski, Algebraic independence and measure, Fund. Math. 61 (1967), 165-169. MR 0224762 (37:361)
  • [7] W. M. Schmidt, On badly approximable numbers and certain games, Trans. Amer. Math. Soc. 123 (1966), 178-199. MR 0195595 (33:3793)
  • [8] -, Diophantine approximation, Lecture Notes in Math., vol. 785, Springer-Verlag, Berlin and New York, 1980. MR 568710 (81j:10038)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0701515-3
Article copyright: © Copyright 1983 American Mathematical Society

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