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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Effective lower bounds for some linear forms

Author: T. W. Cusick
Journal: Trans. Amer. Math. Soc. 222 (1976), 289-301
MSC: Primary 10F35
MathSciNet review: 0422173
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Abstract: It is proved that if $ 1, \alpha ,\beta $ are numbers, linearly independent over the rationals, in a real cubic number field, then given any real number $ d \geqslant 2$, for any integers $ {x_0},{x_1},{x_2}$ such that $ \vert{\text{norm}}({x_0} + \alpha {x_1} + \beta {x_2})\vert \leqslant d$, there exist effectively computable numbers $ c > 0$ and $ k > 0$ depending only on $ \alpha $ and $ \beta $ such that $ \vert{x_1}{x_2}\vert{(\log \vert{x_1}{x_2}\vert)^{k\log d}}\vert{x_0} + \alpha {x_1} + \beta {x_2}\vert > c$ holds whenever $ {x_1}{x_2} \ne 0$. It would be of much interest to remove the dependence on d in the exponent of $ \log \vert{x_1}{x_2}\vert$, for then, among other things, one could deduce, for cubic irrationals, a stronger and effective form of Roth's Theorem.

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Keywords: Diophantine approximation, Roth's Theorem, real cubic fields, Baker's effective estimates
Article copyright: © Copyright 1976 American Mathematical Society