Gaps theorems for linear forms with cubic coefficients

Abstract

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 $\mathbf{\tau}\in\mathbb{N}^2$.