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Mathematics of Computation

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Diophantine approximation of ternary linear forms. II

Author: T. W. Cusick
Journal: Math. Comp. 26 (1972), 977-993
MSC: Primary 10F15
MathSciNet review: 0321879
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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}\vert x + \theta y + {\theta ^2}z\vert$, where the min is taken over all integers $ x,y,z$ satisfying $ 1 \leqq \max (\vert y\vert,\vert z\vert) \leqq M$. Its value is $ (2\theta + 3)/7 \approx .78485$. The same method can be applied to other constants of the same type.

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Keywords: Ternary linear forms, Dirichlet's Diophantine approximation theorem, totally real cubic field
Article copyright: © Copyright 1972 American Mathematical Society

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