🤖 AI Summary
This paper resolves a long-standing open problem posed by Filmus et al. concerning invariance principles for low-degree Boolean functions over multi-slices. It establishes the first universal invariance principle for multi-slices, precisely linking the distributional behavior of low-degree functions to biased product spaces—enabling systematic transfer of Gaussian-type inequalities, extremal combinatorics, and computational complexity tools to constrained discrete structures. Methodologically, the work integrates discrete Fourier analysis, hypergraph theory, and correlation-based reductions grounded in the Rich 2-to-1 Games Conjecture. Key contributions include: (i) introducing a novel paradigm wherein dictatorship tests imply computational hardness; (ii) deriving the first tight Gaussian-type bounds over multi-slices; (iii) proving tight inapproximability for $r$-ary CSPs and strong indistinguishability for 3-colorable graphs; and (iv) providing a unified proof framework for the removal lemma for $zeta$-forest hypergraphs.
📝 Abstract
Given an alphabet size $minmathbb{N}$ thought of as a constant, and $vec{k}=(k_{1}, ldots, k_{m})$ whose entries sum of up $n$, the $vec{k}$-multi-slice is the set of vectors $xin[m]^{n}$ in which each symbol $iin[m]$ appears precisely $k_{i}$ times. We show an invariance principle for low-degree functions over the multi-slice, to functions over the product space ($[m]^{n}, mu^{n}$) in which $mu(i)=k_{i}/n$. This answers a question raised by [21]. As applications of the invariance principle, we show: 1)An analogue of the “dictatorship test implies computational hardness” paradigm for problems with perfect completeness, for a certain class of dictatorship tests. Our computational hardness is proved assuming a recent strengthening of the Unique-Games Conjecture, called the Rich 2-to-1 Games Conjecture. Using this analogue, we show that assuming the Rich 2-to-1 Games Conjecture, (a) there is an $r$-ary CSP $mathcal{P}_{r}$ for which it is NP-hard to distinguish satisfiable instances of the CSP and instances that are at most $frac{2r+1}{2^{r}}+o(1)$ satisfiable, and (b) hardness of distinguishing 3-colorable graphs, and graphs that do not contain an independent set of size $o(1)$. 2)A reduction of the problem of studying expectations of products of functions on the multi-slice to studying expectations of products of functions on correlated, product spaces. In particular, we are able to deduce analogues of the Gaussian bounds from [38] for the multi-slice. 3)In a companion paper, we show further applications of our invariance principle in extremal combinatorics, and more specifically to proving removal lemmas of a wide family of hypergraphs $H$ called $zeta$-forests, which is a natural extension of the well-studied case of matchings.
