Kronecker序列超越环面：最近邻距离与最佳返回
Kronecker sequences beyond the torus: nearest-neighbour distances and best returns
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中文总结 AI 辅助
本文将三间隙定理推广至混合商空间，证明最近邻距离界由实因子控制，并给出最佳返回分母的递推性质及其在超度量因子下的失效与堆积递推的保持。
中文摘要 AI 辅助
经典的三间隙定理指出，圆上的有限Kronecker序列至多有三种间隙长度。我们将这一现象推广到商空间$(V\times U)/\Lambda$上的最近邻距离，其中$V$是有限维实赋范空间，$U$是任意超度量阿贝尔群，且$\Lambda$是闭子群。对于由最大乘积度量诱导的商度量，一致距离界完全由实因子控制。包含在子群中的连续实方向可以被分解出来；对于内积度量，仅由投影子群生成的实方向起作用。对于$\mathbb{R}^d$上的最大范数，普适混合界为$2^d+1$。这些结果统一并推广了Chevallier、Haynes--Ramirez、Das--Haynes和Shulga的界。最佳返回分母的行为有所不同。我们证明，在任意具有平移不变度量的阿贝尔群中，递推关系$q_{n+M}\ge q_n+q_{n+1}$蕴含至多$M+1$个最近邻距离。Shulga的递推从紧实环面推广到任意纯实商空间，其指数由子群的离散部分的秩决定，而非由环境维度决定。然而，在添加超度量因子后，该递推可能失效，即使在adelic环面上也是如此。尽管如此，对于任意混合商空间，堆积递推仍然成立，并且在有效欧几里得维度一和二中，相对于纯实递推的一步损失是精确的。
英文摘要
The classical three-gap theorem says that a finite Kronecker sequence on the circle has at most three gap lengths. We extend this phenomenon to nearest-neighbour distances on quotients $(V\times U)/Λ$, where $V$ is a finite-dimensional real normed space, $U$ is an arbitrary ultrametric abelian group and $Λ$ is closed. For the quotient metric induced by the maximum product metric, the uniform distance bound is controlled entirely by the real factor. Continuous real directions contained in the subgroup can be factored out; for inner-product metrics, only the real directions generated by the projected subgroup matter. For the maximum norm on $\mathbb{R}^d$ the universal mixed bound is $2^d+1$. These results unify and extend bounds of Chevallier, Haynes--Ramirez, Das--Haynes and Shulga. Best-return denominators behave differently. We show that a recurrence $q_{n+M}\ge q_n+q_{n+1}$ implies at most $M+1$ nearest-neighbour distances in any abelian group with a translation-invariant metric. Shulga's recurrence extends from compact real tori to arbitrary purely real quotients, with the index determined by the rank of the discrete part of the subgroup rather than by the ambient dimension. After adding an ultrametric factor, however, this recurrence can fail, even on an adelic solenoid. Nevertheless a packing recurrence survives for arbitrary mixed quotients, and in effective Euclidean dimensions one and two the one-step loss from the purely real recurrence is sharp.
发表机构
- University of York（约克大学）
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