Let $X$ be a $d$-dimensional simplicial complex. A function $F\colon X(k)\to \{0,1\}^k$ is said to be a direct product function if there exists a function $f\colon X(1)\to \{0,1\}$ such that $F(\sigma) = (f(\sigma_1), \ldots, f(\sigma_k))$ for each $k$-face $\sigma$. In an effort to simplify components of the PCP theorem, Goldreich and Safra introduced the problem of direct product testing, which asks whether one can test if $F\colon X(k)\to \{0,1\}^k$ is correlated with a direct product function by querying $F$ on only $2$ inputs. Dinur and Kaufman conjectured that there exist bounded degree complexes with a direct product test in the small soundness regime. We resolve their conjecture by showing that for all $\delta>0$, there exists a family of high-dimensional expanders with degree $O_{\delta}(1)$ and a $2$-query direct product tester with soundness $\delta$. We use the characterization given by a subset of the authors and independently by Dikstein and Dinur, who showed that some form of non-Abelian coboundary expansion (which they called "Unique-Games coboundary expansion") is a necessary and sufficient condition for a complex to admit such direct product testers. Our main technical contribution is a general technique for showing coboundary expansion of complexes with coefficients in a non-Abelian group. This allows us to prove that the high dimensional expanders constructed by Chapman and Lubotzky satisfies the necessary conditions, thus admitting a 2-query direct product tester with small soundness.

翻译：设 $X$ 为 $d$ 维单纯复形。若存在函数 $f\colon X(1)\to \{0,1\}$ 使得对每个 $k$ 维面 $\sigma$ 均有 $F(\sigma) = (f(\sigma_1), \ldots, f(\sigma_k))$，则称函数 $F\colon X(k)\to \{0,1\}^k$ 为直接积函数。为简化PCP定理的组成部分，Goldreich与Safra提出了直接积测试问题：能否仅通过查询 $F$ 的 $2$ 个输入来判定 $F\colon X(k)\to \{0,1\}^k$ 是否与某个直接积函数相关？Dinur与Kaufman猜想：在小可靠性条件下，存在具有有界度的复形满足直接积测试性质。我们通过证明对任意 $\delta>0$，存在度为 $O_{\delta}(1)$ 的高维扩展复形族及可靠度为 $\delta$ 的 $2$-查询直接积测试器来解决该猜想。本文利用部分作者独立与Dikstein-Dinur给出的刻画：某种形式的非阿贝尔上同调扩张（称为"唯一博弈上同调扩张"）是复形允许此类直接积测试器的充要条件。我们的主要技术贡献是提出了一种处理非阿贝尔群系数复形上同调扩张的通用方法。由此证明Chapman与Lubotzky构造的高维扩展复形满足必要条件，从而允许小可靠性的 $2$-查询直接积测试器。