AI 中文总结
本文证明三次数域中线性形式小数部分间隙个数随参数无界，并给出复嵌入情形下带对数下界的显式构造。
AI 中文摘要
设 $1,\alpha_1,\alpha_2$ 是三次数域 $K\subset\mathbb{R}$ 的一组基，对于 $\tau_1,\tau_2\ge 1$，令 $G(\mathbf{\alpha},\mathbf{\tau})$ 表示数 $m_1\alpha_1+m_2\alpha_2$（其中 $m_1,m_2\in\mathbb{Z}$，$0\le m_1<\tau_1$，$0\le m_2<\tau_2$）的小数部分之间不同间隙的个数。我们证明当 $\mathbf{\tau}$ 变化时 $G(\mathbf{\alpha},\mathbf{\tau})$ 无界。当 $K$ 为全实数域时，这可由 Shapira 关于格空间上对角群轨道的密度定理推出。当 $K$ 具有复嵌入时，我们给出一个显式构造，该构造还表明存在常数 $c,\kappa>0$，使得对无穷多个 $\mathbf{\tau}\in\mathbb{N}^2$，有 $G(\mathbf{\alpha},\mathbf{\tau})\ge c(\log\max\{\tau_1,\tau_2\})^{\kappa}$。
英文摘要
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 $m_1,m_2\in\mathbb{Z}$, $0\le m_1<τ_1$ and $0\le m_2<τ_2$. We show that $G(\mathbfα,\mathbfτ)$ is unbounded as $\mathbfτ$ 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 $c,κ>0$ for which $G(\mathbfα,\mathbfτ)\ge c(\log\max\{τ_1,τ_2\})^κ$ for infinitely many $\mathbfτ\in\mathbb{N}^2$.
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