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Diophantine approximation of ternary linear forms. (English) Zbl 0217.03803

The paper gives an efficient method for finding arbitrarily many solutions in integers \(x,y,z\) of the Diophantine inequality \[ \vert x+\alpha y + \beta z\vert \max(y^2, z^2 ) < c, \] where the numbers \(1,\alpha,\beta\) form an integral basis for a totally real cubic field over the rationals. If \(c\) is small enough, the method generates all solutions of the inequality. The algorithm introduced in the paper has features similar to those of the simple continued fraction algorithm for a single real number, and so can be used to obtain detailed information about the solutions of the Diophantine inequality under consideration.
For example, in another paper [“Diophantine approximation of ternary linear forms. II”. Math. Comput. 26, 977–993 (1972; Zbl 0258.10016)] the algorithm is used in the exact evaluation of the constant \(\limsup_{N\to\infty} \min M^2 \vert x + \theta y + \theta^2 z\vert\), where the \(\min\) is taken over all integers \(x,y,z\) satisfying \(1\le \max(\vert y\vert,\vert z\vert)\le M\) and where \(\theta\) denotes the positive root of the equation \(x^3 + x^2 - 2x - 1= 0\), that is, \(\theta = 2 \cos(2\pi/7)\); the value of the constant is \((2\theta + 3)/7 \approx .78485\).
The interest of constants of this type in the theory of Diophantine approximation is discussed in a paper of H. Davenport and W. M. Schmidt [Symp. Math. 4, 113–132 (1970; Zbl 0226.10032)], where estimates for the particular constant just mentioned are given. The method of the paper under review can be extended to linear forms in \(n\) variables with coefficients \(1, \alpha_1, \ldots, \alpha_{n-1}\) which form a basis (not necessarily integral) for a real (not necessarily totally real) number field of degree \(n\).

MSC:

11J25 Diophantine inequalities
11J13 Simultaneous homogeneous approximation, linear forms
11Y50 Computer solution of Diophantine equations
Full Text: DOI

References:

[1] J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957. · Zbl 0077.04801
[2] H. Hancock, Development of the Minkowski Geometry of Numbers, Macmillan, New York, 1939, pp. 371-452. MR 1, 67. · Zbl 0060.11206
[3] H. Minkowski, ”Zur Theorie der Kettenbrüche,” in Gesammelte Abhandlungen. Vol. I, Teubner, Leipzig, 1911, pp. 278-292.
[4] L. G. Peck, Simultaneous rational approximations to algebraic numbers, Bull. Amer. Math. Soc. 67 (1961), 197 – 201. · Zbl 0098.26302
[5] M. Zeisel, ”Zur Minkowskischen Parallelepipedapproximation,” S.-B. Akad. Wiss. Wien, v. 126, 1917, pp. 1221-1247.
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