Diophantine approximation of ternary linear forms. II
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- by T. W. Cusick PDF
- Math. Comp. 26 (1972), 977-993 Request permission
Abstract:
Let $\theta$ denote the positive root of the equation ${x^3} + {x^2} - 2x - 1 = 0$; that is, $\theta = 2\cos (2\pi /7)$. The main result of the paper is the evaluation of the constant $\lim {\sup _{M \to \infty }}\min {M^2}|x + \theta y + {\theta ^2}z|$, where the min is taken over all integers $x,y,z$ satisfying $1 \leqq \max (|y|,|z|) \leqq M$. Its value is $(2\theta + 3)/7 \approx .78485$. The same method can be applied to other constants of the same type.References
- J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957. MR 0087708
- T. W. Cusick, Diophantine approximation of ternary linear forms, Math. Comp. 25 (1971), 163–180. MR 296022, DOI 10.1090/S0025-5718-1971-0296022-4
- H. Davenport and Wolfgang M. Schmidt, Dirichlet’s theorem on diophantine approximation, Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69) Academic Press, London, 1970, pp. 113–132. MR 0272722
- H. Davenport and W. M. Schmidt, Dirichlet’s theorem on diophantine approximation. II, Acta Arith. 16 (1969/70), 413–424. MR 279040, DOI 10.4064/aa-16-4-413-424 V. Jarnik, “Problem 278,” Colloq. Math., v. 6, 1958, pp. 337-338. J. Lesca, Thesis, University of Grenoble, France, 1968.
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Math. Comp. 26 (1972), 977-993
- MSC: Primary 10F15
- DOI: https://doi.org/10.1090/S0025-5718-1972-0321879-9
- MathSciNet review: 0321879