九距离定理与最佳逼近分母的增长
Nine-distance theorem and growth of best-approximation denominators
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中文总结 AI 辅助
本文证明平坦三维环面Kronecker序列的九距离定理，证实Haynes与Marklof的猜想，给出d维环面的距离上界，核心是最佳同时逼近分母的新增长定理。
中文摘要 AI 辅助
我们证明了平坦三维环面上Kronecker序列的九距离定理，即在前N个轨道点中，最多出现9个不同的正最近邻距离，这证明了Haynes和Marklof的猜想，Dettmann的例子表明9是最优的。更一般地，我们证明在平坦d维环面上此类距离的数量最多为2^d+1。主要工具是d维内积空间中最佳同时逼近分母q₁<q₂<…的新增长定理，其本身具有独立意义。我们证明，只要q_{n+2^d}有定义，要么q_{n+2^d}≥2q_{n+1}，要么可将1,…,2^d划分为不相交对{j,k}（j<k），使得q_{n+k}=q_n+q_{n+j}。特别地，q_{n+2^d}≥min{2q_{n+1},q_n+q_{n+2^{d-1}}}≥q_n+q_{n+1}。
英文摘要
We prove a nine-distance theorem for Kronecker sequences on flat three-tori. That is, we show that among the first $N$ orbit points, at most nine distinct positive nearest-neighbour distances occur. This proves the conjecture of Haynes and Marklof. An example of Dettmann shows that nine is optimal. More generally, we prove that on a flat $d$-dimensional torus the number of such distances is at most $2^d+1$. The main tool is a new growth theorem for the denominators $q_1<q_2<\cdots$ of best simultaneous approximations in a $d$-dimensional inner-product space, which is of independent interest. We prove that, whenever $q_{n+2^d}$ is defined, either $q_{n+2^d}\ge2q_{n+1}$, or the indices $1,\ldots,2^d$ can be partitioned into disjoint pairs $\{j,k\}$, $j<k$, such that $q_{n+k}=q_n+q_{n+j}$. In particular, $$ q_{n+2^d}\ge \min\{2q_{n+1},q_n+q_{n+2^{d-1}}\}\ge q_n+q_{n+1}. $$
