J Austral Math Soc Ser A 46 pp236--250, 1989.

The Asymmetric Product of Three Inhomogenous Linear Formss

V. K. Grover

Abstract

Let L be a lattice in R₃ of determinant 1. Define the homogenous minimum of L as m_h(L) = inf |u₁u₂u₃| extended over all points (u₁, u₂, u₃) of L other than the orogin. It is shown that for any given (c₁, c₂, c₃) in R₃ there exists a point (u₁, u₂, u₃) of L for which

-r £ (u1 + c1)(u2 + c2)(u3 + c3) £ s,      r, s > 0,

provided that rs > 1/64 if m_h(L) = 0, and rs ³ 1/16.81 if m_h(L) > 0.

1980 AMS Subject Classification (1985 Revision): 10E15

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Authors

V. K. Grover

Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-160 014, India.