Minima of a decomposable cubic form of three variables

There is given a proof of the theorem, asserting that if for all $X\in\mathbb{Z}^3$ ($X\ne0$) we have $|F(X)|\geq m>0$, 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 $\alpha,\beta\in\mathbb{R}$ such that $q\| q\alpha\|\cdot\|q\beta\|>x>0$ for all natural numbers $q$? From the result of the paper there follows that such $(\alpha,\beta)$ do not exist.