Diophantine approximation of ternary linear forms. II

Cusick, T. W.

doi:10.1090/S0025-5718-1972-0321879-9

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

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