Upper Bounds for Symmetric Approximate Bounded Indistinguishability

A pair of probability distributions over $\{0,1\}^n$ is said to be $(k,\delta)$-wise indistinguishable if all of the size $k$ marginals are within statistical distance at most $\delta$. 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 $\Omega(1)$. 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 $\delta$ for any $\delta\leq\exp\left({-\omega(\sqrt{n\log{n}})}\right)$. Our bounds also yield new results in other regimes, for example when $k$ is sublinear or when $t/n$ tends to 1.