AI 中文总结
该研究引入序列失真谱，定义了\(f -\)失真序列，探讨其可实现性与度量空间类型及\(f\)增长类型的关系，分类了欧氏空间等不同空间中幂次速率的情况，还构造了两对有特定性质的恰当测地空间，用序列失真区分它们。
AI 中文摘要
我们引入了“序列失真谱”，它是一种拟等距不变量，记录了在度量空间中由自然数集或整数集索引的序列的大规模距离分布。对于速率函数\(f:\mathbb N\to(0,\infty)\)（当整数\(t\leq0\)时扩展为\(f(t)=0\)），度量空间\(X\)中的序列\((p_n)\)若存在整数\(C\geq1\)，使得对所有\(m,n\)有\(\frac{1}{C}f(\lfloor \frac{1}{C}|n - m|-C\rfloor)\leq d(p_n,p_m)\leq C f(C|n - m|+C)+C\)，则称该序列是“\(f\)-失真的”。此定义意味着\(f(N)=O(N)\)。对于速率函数，可实现性仅取决于环境拟等距类型和\(f\)的增长类型。我们对欧几里得空间中可能的幂次速率\(f(N)=N^\alpha\)（\(0\lt\alpha\leq1\)）进行了分类：在\(\mathbb R\)中仅出现线性速率\(\alpha = 1\)，而在\(\mathbb R^k\)（\(k\geq2\)）中，可实现的指数恰好是\(1/k\lt\alpha\leq1\)。对于测地\(\delta -\)双曲空间\(X\)，不存在\(0\lt\alpha\lt1\)的幂次速率\(N^\alpha\)。双曲平面也实现了对数速率。\(X\)的指数填充界排除了每个\(o(\log N)\)速率，但具有无界几何的恰当CAT\((-1)\)曲面实现了对数 - 对数速率。在任意单纯树中，每个可实现的速率在常数范围内都是线性的。最后，我们构造了两对恰当测地空间：第一对具有等价的基点填充函数，第二对具有等价的一致填充函数；两对都具有相等的渐近维数和填充函数增长类，并且对于每个公共缩放序列和超滤子，在所选楔点处具有等距渐近锥。然而，序列失真区分了每一对，并且第二对具有有界几何。
英文摘要
We introduce \emph{sequence distortion spectrum}, a quasi-isometry invariant recording the large-scale distance profiles of sequences indexed by $\mathbb N$ or $\mathbb Z$ in a metric space. For a rate function $f:\mathbb N\to(0,\infty)$, extended by $f(t)=0$ for integers $t\le 0$, a sequence $(p_n)$ in a metric space $X$ is \emph{$f$-distorted} if there exists an integer $C\ge 1$ such that for all $m,n$ we have $$\frac{1}{C}f(\lfloor \frac{1}{C}|n-m|-C\rfloor)\le d(p_n,p_m)\le C f(C|n-m|+C)+C.$$ This definition implies that $f(N)=O(N)$. For rate functions, realizability depends only on the ambient quasi-isometry type and the growth type of $f$. We classify the possible power rates $f(N)=N^α$ (where $0<α\le 1)$ for Euclidean spaces: in $\mathbb R$ only the linear rate $α=1$ occurs, while in $\mathbb R^k$, $k\ge2$, the realizable exponents are exactly $1/k<α\le 1$. For a geodesic $δ$-hyperbolic space $X$, no power rate $N^α$ with $0<α<1$ occurs. The hyperbolic plane also realizes the logarithmic rate. An exponential packing bound for $X$ rules out every $o(\log N)$ rate, but a proper CAT$(-1)$ surface of unbounded geometry realizes a log--log rate. In an arbitrary simplicial tree, every realizable rate is linear up to constants. Finally, we construct two pairs of proper geodesic spaces: the first has equivalent basepoint packing functions and the second equivalent uniform packing functions; both pairs have equal asymptotic dimensions and filling-function growth classes, and isometric asymptotic cones at the chosen wedge points for every common scaling sequence and ultrafilter. Yet sequence distortion distinguishes each pair, and the second pair has bounded geometry.
Comments28 pages
