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 $\{f_s:s\to\Sigma\}_{s\in X}$ be an ensemble of local functions, each defined over a subset $s\subset [n]$. Suppose that we run the following so-called agreement test: choose a random pair of sets $s_1,s_2\in X$ that intersect on $\sqrt k$ elements, and accept if $f_{s_1},f_{s_2}$ agree on the elements in $s_1\cap s_2$. We denote the success probability of this test by $Agr(\{f_s\})$. Given that $Agr(\{f_s\})=\epsilon>0$, is there a global function $G:[n]\to\Sigma$ such that $f_s = G|_s$ for a non-negligible fraction of $s\in X$ ? 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, $Agr (\{f_s\}) > \epsilon \; \; \longrightarrow$ there is a function $G:[n]\to\Sigma$ such that $\Pr_s[f_s\overset{0.99}{\approx} G|_s]\geq poly(\epsilon)$. 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.

翻译：我们通过构造不具有小连通覆盖的新型高维扩展器，解决了低接受条件下的去随机化直接乘积测试问题。研究表明，我们的复形具有交换上循环扩展性质，这使得我们能够基于先前工作推导出一致性定理。去随机化直接乘积测试（亦称一致性测试）涉及以下问题：设X是[n]的k元子集族，$\{f_s:s\to\Sigma\}_{s\in X}$是定义在子集$s\subset [n]$上的局部函数系综。假设执行如下所谓的一致性测试：随机选取一对在$\sqrt k$个元素上相交的集合$s_1,s_2\in X$，若$f_{s_1},f_{s_2}$在$s_1\cap s_2$上一致则接受。记该测试的成功概率为$Agr(\{f_s\})$。给定$Agr(\{f_s\})=\epsilon>0$，是否存在全局函数$G:[n]\to\Sigma$，使得对于不可忽略比例的$s\in X$有$f_s = G|_s$？我们构造了[n]的k子集族X，满足$|X| = O(n)$且支持低接受一致性定理。即，当$Agr (\{f_s\}) > \epsilon$时，存在函数$G:[n]\to\Sigma$使得$\Pr_s[f_s\overset{0.99}{\approx} G|_s]\geq poly(\epsilon)$。关键思想是用辛高维扩展器（HDXs）替代已得到充分研究的LSV复形。该族X正是新型辛HDXs的k维面。后者由于基本群满足同余子群性质（意味着缺乏小覆盖），能更好地满足我们的需求。