Diophantine approximation of ternary linear forms

Abstract

The paper gives an efficient method for finding arbitrarily many solutions in integers x, y, z of the Diophantine inequality | x + α y + β z | max ( y 2 , z 2 ) > c |x + \alpha y + \beta z|\max ({y^2},{z^2}) > c , where α \alpha defines a totally real cubic field F over the rationals, the numbers 1, α , β \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.