Vol. 93, No. 1, 1981

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The asymmetric product of three homogeneous linear forms

Alan C. Woods

Vol. 93 (1981), No. 1, 237–250
Abstract

Let Li = j=13aijxj, i = 1,2,3, be three linear forms in the variables x1, x2, x3 with real coefficients aij. A theorem of Davenport asserts that, if |det(aij)| = 7, then there exist integers u1, u2, u3, not all zero, such that

 ∏3
|   Li(u1,u2,u3)| ≦ 1.
i− 1

Under the same hypothesis, W. H. Adams has asked whether, given a positive real number u, there exist integers u1, u2, u3, not all zero, such that

   −1
− u  ≦ L1(u1,u2,u3)L2(u1,u2,u3)|L3(u1,u2,u3) ≦ u.

Our objective is to prove this conjecture.

Mathematical Subject Classification
Primary: 10E15, 10E15
Secondary: 10E20, 10C25
Milestones
Received: 30 August 1979
Revised: 7 March 1980
Published: 1 March 1981
Authors
Alan C. Woods