## 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

- A. Baker,
*Effective methods in Diophantine problems*, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 195–205. MR**0314774** - A. Baker,
*A sharpening of the bounds for linear forms in logarithms*, Acta Arith.**21**(1972), 117–129. MR**302573**, DOI 10.4064/aa-21-1-117-129 - A. Baker,
*Effective methods in Diophantine problems. II*, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 1–7. MR**0337802** - A. I. Borevich and I. R. Shafarevich,
*Number theory*, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR**0195803** - J. W. S. Cassels,
*An introduction to Diophantine approximation*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957. MR**0087708** - J. W. S. Cassels and H. P. F. Swinnerton-Dyer,
*On the product of three homogeneous linear forms and the indefinite ternary quadratic forms*, Philos. Trans. Roy. Soc. London Ser. A**248**(1955), 73–96. MR**70653**, DOI 10.1098/rsta.1955.0010 - T. W. Cusick,
*Diophantine approximation of linear forms over an algebraic number field*, Mathematika**20**(1973), 16–23. MR**340186**, DOI 10.1112/S0025579300003582 - P. Gallagher,
*Metric simultaneous diophantine approximation*, J. London Math. Soc.**37**(1962), 387–390. MR**157939**, DOI 10.1112/jlms/s1-37.1.387 - Kurt Mahler,
*Ein Übertragungsprinzip für lineare Ungleichungen*, Časopis Pěst. Mat. Fys.**68**(1939), 85–92 (German). MR**0001241** - L. G. Peck,
*Simultaneous rational approximations to algebraic numbers*, Bull. Amer. Math. Soc.**67**(1961), 197–201. MR**122772**, DOI 10.1090/S0002-9904-1961-10565-X - K. F. Roth,
*Rational approximations to algebraic numbers*, Mathematika**2**(1955), 1–20; corrigendum, 168. MR**72182**, DOI 10.1112/S0025579300000644 - Wolfgang M. Schmidt,
*Simultaneous approximation to algebraic numbers by rationals*, Acta Math.**125**(1970), 189–201. MR**268129**, DOI 10.1007/BF02392334 - D. C. Spencer,
*The lattice points of tetrahedra*, J. Math. Phys. Mass. Inst. Tech.**21**(1942), 189–197. MR**7767**, DOI 10.1002/sapm1942211189

## 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