凸范数的五距离定理
The Five Distance Theorem For An Arbitrary Norm
AI总结：
该论文证明了二维严格凸范数下，幺模格上克罗内克序列的最近邻距离数至多为五，解决了对应ℓₚ范数的猜想界，扩展了相关格论论证。
AI中文摘要：
三间隙定理指出，克罗内克序列α,2α,…,Nα（模1）将圆划分为至多三种不同长度的区间。在二维最近邻类比中，Haynes和Marklof证明，任意幺模格上的克罗内克序列（模该格）在欧氏范数下确定至多五种不同的最近邻距离，且该界是紧的。随后Dettmann构造了例子，使得对所有ℓₚ范数（1≤p≤∞）都达到五种不同距离。我们证明了ℝ²上所有严格凸范数对应的上界：对任意幺模格L、任意α∈ℝ²及任意自然数N，不同最近邻距离的数量至多为五。特别地，这解决了所有1<p<∞的ℓₚ范数的猜想界，该界是最优的。证明通过用基于适当Brass角测度的锥引理替代Haynes和Marklof的欧氏角估计，扩展了其格论论证。
英文摘要：
The three gap theorem states that the points of the Kronecker sequence $α,2α,\ldots,Nα$, considered modulo one, divide the circle into intervals of at most three distinct lengths. In a two-dimensional nearest-neighbour analogue, Haynes and Marklof proved that the Kronecker sequence modulo an arbitrary unimodular lattice determines at most five distinct nearest-neighbour distances in the Euclidean norm, and that this bound is sharp. Dettmann subsequently constructed examples attaining five distinct distances for every $\ell_p$-norm, $1\leq p\leq\infty$. We prove the corresponding upper bound for every norm on $\mathbb R^2$: for every full-rank lattice $L$, every $\boldsymbolα\in\mathbb R^2$, and every $N\in\mathbb N$, the number of distinct nearest-neighbour distances is at most five. For strictly convex norms, the proof extends the lattice-theoretic argument of Haynes and Marklof by replacing the Euclidean angular estimates with a cone lemma based on a proper Brass angular measure. The result for arbitrary norms is then obtained by a strictly convex perturbation and a limiting argument.
