Effective lower bounds for some linear forms
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- by T. W. Cusick PDF
- Trans. Amer. Math. Soc. 222 (1976), 289-301 Request permission
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 $|{\text {norm}}({x_0} + \alpha {x_1} + \beta {x_2})| \leqslant d$, there exist effectively computable numbers $c > 0$ and $k > 0$ depending only on $\alpha$ and $\beta$ such that $|{x_1}{x_2}|{(\log |{x_1}{x_2}|)^{k\log d}}|{x_0} + \alpha {x_1} + \beta {x_2}| > c$ holds whenever ${x_1}{x_2} \ne 0$. It would be of much interest to remove the dependence on d in the exponent of $\log |{x_1}{x_2}|$, for then, among other things, one could deduce, for cubic irrationals, a stronger and effective form of Roth’s Theorem.References
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Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 222 (1976), 289-301
- MSC: Primary 10F35
- DOI: https://doi.org/10.1090/S0002-9947-1976-0422173-8
- MathSciNet review: 0422173