Low Acceptance Agreement Tests via Bounded-Degree Symplectic HDXs

We solve the derandomized direct product testing question in the low acceptance regime, by constructing new high dimensional expanders that have no small connected covers. We show that our complexes have swap cocycle expansion, which allows us to deduce the agreement theorem by relying on previous work. Derandomized direct product testing, also known as agreement testing, is the following problem. Let X be a family of k-element subsets of [n] and let {f_s:s→Σ}_s∈ X be an ensemble of local functions, each defined over a subset s⊂ [n]. Suppose that we run the following so-called agreement test: choose a random pair of sets s_1,s_2∈ X that intersect on √(k) elements, and accept if f_s_1,f_s_2 agree on the elements in s_1∩ s_2. We denote the success probability of this test by Agr({f_s}). Given that Agr({f_s})=ϵ>0, is there a global function G:[n]→Σ such that f_s = G|_s for a non-negligible fraction of s∈ X ? We construct a family X of k-subsets of [n] such that |X| = O(n) and such that it satisfies the low acceptance agreement theorem. Namely, Agr ({f_s}) > ϵ ⟶ there is a function G:[n]→Σ such that _s[f_s0.99≈ G|_s]≥ poly(ϵ). A key idea is to replace the well-studied LSV complexes by symplectic high dimensional expanders (HDXs). The family X is just the k-faces of the new symplectic HDXs. The later serve our needs better since their fundamental group satisfies the congruence subgroup property, which implies that they lack small covers.