排序扩散度与基于样本的测试
Ranked spreadness and sample-based testing
AI总结：
该研究提出排序扩散度概念，证明相关集合系统的无宽度击中、加权集中及核提取定理，并将其应用于基于样本的测试器模拟非自适应属性测试器，得到与已有猜想匹配的样本复杂度指数结果。
AI中文摘要：
在这篇短文中，我们引入了排序扩散度（ranked spreadness）的概念，这是对常规扩散条件的强化：在该条件下，每个集合的元素可被排序，使得它们的单坐标边缘分布随秩呈几何衰减。这种额外结构消除了随机包含估计中对最大集合规模的依赖。我们证明了排序扩散集合系统的无宽度击中定理与加权集中定理，还给出了一个基础的核提取定理，表明排序扩散度会自然出现在小集合的任意分布中。\n我们的主要应用是用基于样本的测试器模拟非自适应属性测试器。若一个单边测试器的平均查询复杂度为$d$，且以至少$δ$的概率拒绝所有远离（目标属性）的输入，那么对任意整数$c>d/δ$，都存在一个单边的基于样本的模拟方案，其期望样本复杂度为$O_{d,δ,|Σ|}\igl(n^{1-1/c}\igr)$。更一般地，若正输入被拒绝的概率至多为$γ$，远离输入被拒绝的概率至少为$δ>γ$，则对所有$c>d/(δ-γ)$，上述结论依然成立。特别地，对于常数查询的非自适应测试器，我们得到了指数为$1-Θ(1/q)$的结果，这与Fischer、Lachish和Vasudev猜想的指数（在拒绝间隙的依赖项范围内）相匹配。
英文摘要：
In this note, we introduce the notion of ranked spreadness, a strengthening of the usual spread condition in which the elements of each member can be ordered so that their one-coordinate marginals decay geometrically with their rank. This additional structure removes the dependence on the maximum set size in random-containment estimates. We prove width-free hitting and weighted-concentration theorems for ranked-spread set systems, together with an elementary kernel-extraction theorem showing that ranked spreadness arises naturally in arbitrary distributions on small sets. Our main application is to the simulation of nonadaptive property testers by sample-based testers. If a one-sided tester has average query complexity $d$ and rejects every far input with probability at least $δ$, then, for every integer $c>d/δ$, it admits a one-sided sample-based simulation with expected sample complexity $O_{d,δ,|Σ|}\bigl(n^{1-1/c}\bigr)$. More generally, if positive inputs are rejected with probability at most $γ$ and far inputs with probability at least $δ>γ$, the same conclusion holds for every $c>d/(δ-γ)$. In particular, for constant-query nonadaptive testers we obtain an exponent $1-Θ(1/q)$, matching, up to the dependence on the rejection gap, the exponent conjectured by Fischer, Lachish, and Vasudev.
