Minimum of a decomposable cubic form of three variables

Abstract

There is given a proof of the theorem, asserting that if for all X∈ℤ3(X≠0) we have ∥F(X)∥≥m>0, where F(X) is a decomposable cubic form of three variables, then F(X) is proportional to an integral form. Making use of this result, the author gives a proof to Littlewood's problem: can one find α, β ∈ ℝ such that q‖qα‖·‖qβ‖>x>0 for all natural numbers g? From the result of the paper there follows that such (α, β) do not exist.