The asymmetric product of three inhomogeneous linear forms

Let Λ be a lattice in R3 of determinant 1. Define the homogeneous minium of Λ as mn (Λ) = inf |u1, u2, u3| extended over all points (u1, u2, u3) of Λ other than the origin. It is shown that for any given (c1, c2, c3) in R3 there exists a point (u1, u2, u3) of Λ for which provided that ρσ > 1/64 if mn (Λ) = 0, and ρσ ≥1/16.81 if mn (A) > 0.

The asymmetric product of three inhomogeneous linear forms

It is shown, given any positive real number λ and any point (x1, x2, x3) of R3 and any lattice λ R3; that there exists a point (z1, z2, z3) of λ for whichwhich generalizes a theorem due to Remak.

Yet another proof of Minkowski's theorem on the product of two inhomogeneous linear forms

Theorem 1. Let ξ = αx + βy, η = γx + δy be two homogeneous linear forms in x, y with real coefficients and determinant αδ − βγ = Δ ≠ 0. Then for any real constants p, q there are integers x, y such that