Diophantine approximation of ternary linear forms
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- by T. W. Cusick PDF
- Math. Comp. 25 (1971), 163-180 Request permission
Abstract:
The paper gives an efficient method for finding arbitrarily many solutions in integers x, y, z of the Diophantine inequality $|x + \alpha y + \beta z|\max ({y^2},{z^2}) < c$, where $\alpha$ defines a totally real cubic field F over the rationals, the numbers 1, $\alpha ,\beta$ form an integral basis for F, and c is a constant which can be calculated in terms of parameters of the method. For certain values of c, the method generates all solutions of the inequality.References
- 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 H. Hancock, Development of the Minkowski Geometry of Numbers, Macmillan, New York, 1939, pp. 371-452. MR 1, 67. H. Minkowski, "Zur Theorie der Kettenbrüche," in Gesammelte Abhandlungen. Vol. I, Teubner, Leipzig, 1911, pp. 278-292.
- 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 M. Zeisel, "Zur Minkowskischen Parallelepipedapproximation," S.-B. Akad. Wiss. Wien, v. 126, 1917, pp. 1221-1247.
Additional Information
- © Copyright 1971 American Mathematical Society
- Journal: Math. Comp. 25 (1971), 163-180
- MSC: Primary 10F99
- DOI: https://doi.org/10.1090/S0025-5718-1971-0296022-4
- MathSciNet review: 0296022