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|sϵX 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 s₁, s₂ X that intersect on √k elements, and accept if fs₁, f_s2 agree on the elements in s₁∩ s₂. We denote the success probability of this test by Agree f_s Given that Agree (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, Agree (Formula Presented). 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 latter serve our needs better since their fundamental group satisfies the congruence subgroup property, which implies that they lack small covers. We also give a polynomial-time algorithm to construct this family of sym-plectic HDXs.