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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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.
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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