Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 10F35

Retrieve articles in all journals with MSC: 10F35

Additional Information

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