Gaps theorems for linear forms with cubic coefficients

Let $1,{\alpha }_1,{\alpha }_2$ be a basis for a cubic number field $K\subset \mathbb{R}$, and for ${\tau }_1,{\tau }_2\ge 1$ let $G(\mathbf{\alpha },\mathbf{\tau })$ denote the number of distinct gaps between the fractional parts of the numbers $m_1{\alpha }_1+m_2{\alpha }_2$, with $m_1,m_2\in \mathbb{Z}$, $0\le m_1<{\tau }_1$ and $0\le m_2<{\tau }_2$. We show that $G(\mathbf{\alpha },\mathbf{\tau })$ is unbounded as $\mathbf{\tau }$ varies. When $K$ is totally real this is deduced from a density theorem of Shapira for orbits of the diagonal group on the space of lattices. When $K$ has a complex embedding we give an explicit construction, which also shows that there are constants $c,\kappa >0$ for which $G(\mathbf{\alpha },\mathbf{\tau })\ge c(\log \max \{{\tau }_1,{\tau }_2\}{)}^{\kappa }$ for infinitely many $\mathbf{\tau }\in {\mathbb{N}}^2$.