A pair of probability distributions over $\{0,1\}^n$ is said to be $(k,δ)$-wise indistinguishable if all of the size $k$ marginals are within statistical distance at most $δ$. Previous works introduced this concept and study when and how well one can distinguish between such a pair of symmetric distributions by observing $t$ bits. We use a simple hypergeometric smoothing approach and Hahn polynomials to obtain new upper bounds that apply across a wider range of parameters and improve previously available bounds in several regimes. In particular, prior works left open the basic question of whether there exist constants $0<c_1<c_2<1$ and a pair of $(c_1n,0)$-wise indistinguishable distributions such that the $c_2n$-wise marginals have statistical distance $Ω(1)$. One application of our new bounds is to rule this out for all $c_1,c_2$ and to show that the $c_2n$-wise marginals must in fact be exponentially close. Another application in this setting is to show that the $c_2n$-wise marginals must be super-polynomially close even if the $c_1n$-wise marginals are allowed to have statistical distance $δ$ for any $δ\leq\exp\left({-ω(\sqrt{n\log{n}})}\right)$. Our bounds also yield new results in other regimes, for example when $k$ is sublinear or when $t/n$ tends to 1.

翻译：一对定义在$\{0,1\}^n$上的概率分布被称为$(k,δ)$-wise不可区分的，如果所有大小为$k$的边缘分布之间的统计距离至多为$δ$。先前的工作引入了这一概念，并研究了在何种条件下以及如何通过观测$t$个比特来区分这样一对对称分布。我们采用一种简单的超几何平滑方法和哈恩多项式，得到了新的上界，这些上界适用于更广泛的参数范围，并在若干情形下改进了先前已有的界。特别地，先前的工作留下了一个基本未解问题：是否存在常数$0<c_1<c_2<1$以及一对$(c_1n,0)$-wise不可区分的分布，使得其$c_2n$-wise边缘分布的统计距离为$Ω(1)$。我们新上界的一个应用是排除了所有$c_1,c_2$下这一可能性，并证明$c_2n$-wise边缘分布实际上必须指数接近。另一个在该设定中的应用是，即使允许$c_1n$-wise边缘分布对于任意$δ\leq\exp\left({-ω(\sqrt{n\log{n}})}\right)$具有统计距离$δ$，$c_2n$-wise边缘分布也必须是超多项式接近的。我们的界还在其他情形下产生了新结果，例如当$k$为次线性或$t/n$趋近于1时。