Gaps theorems for linear forms with cubic coefficients

Source abstract

Let $1,α_1,α_2$ be a basis for a cubic number field $K\subset\mathbb{R}$, and for $τ_1,τ_2\ge 1$ let $G(\mathbfα,\mathbfτ)$ denote the number of distinct gaps between the fractional parts of the numbers $m_1α_1+m_2α_2$, with $m_1,m_2\in\mathbb{Z}$, $0\le m_1 0$ for which $G(\mathbfα,\mathbfτ)\ge c(\log\max\{τ_1,τ_2\})^κ$ for infinitely many $\mathbfτ\in\mathbb{N}^2$.