论文标题
Lipschitz常量$ \ log {n} $几乎可以肯定足以将$ n $网格点映射到一个立方体
Lipschitz constant $\log{n}$ almost surely suffices for mapping $n$ grid points onto a cube
论文作者
论文摘要
kaluža，kopecká和作者表明，最好的lipschitz是映射的最佳lipschitz，以给定的$ n^{d} $ - 在整数晶格$ \ mathbb {z}^{d} $中设置的元素，以$ n \ in \ mathbb {n} $，$ n $ n $ n $ n $ n $ n $ n $ n $ n \ $ \ left \ {1，\ ldots，n \ right \}^{d} $可能是任意大的。但是，从上方或下面，这种最好的Lipschitz常数如何以$ n $的形式增长，但尚无已知的非平凡渐近渐近界限。我们从概率的角度解决了这个问题。更准确地说，我们考虑给定有限晶格内的$ n^{d} $点的随机配置，并确定，在最佳Lipschitz上，几乎可以肯定的是，$ \ log n $的渐近上限，将其映射的常数用于该集合的常规$ n $ n $ n $ n $ n $ n $ n $ n $ n $ gred $ \ weft \ weft \ weft \ weft \ weft \ weft \ weft \ weft \ weft \ weft \ weft \ weft \ weft \ weft \ weft \ ldots，ndots，ndots，n} \ prirgright。
Kaluža, Kopecká and the author have shown that the best Lipschitz constant for mappings taking a given $n^{d}$-element set in the integer lattice $\mathbb{Z}^{d}$, with $n\in \mathbb{N}$, surjectively to the regular $n$ times $n$ grid $\left\{1,\ldots,n\right\}^{d}$ may be arbitrarily large. However, there remain no known, non-trivial asymptotic bounds, either from above or below, on how this best Lipschitz constant grows with $n$. We approach this problem from a probabilistic point of view. More precisely, we consider the random configuration of $n^{d}$ points inside a given finite lattice and establish almost sure, asymptotic upper bounds of order $\log n$ on the best Lipschitz constant of mappings taking this set surjectively to the regular $n$ times $n$ grid $\left\{1,\ldots,n\right\}^{d}$.
