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 $X$ 是一个 $d$ 维单纯复形。若存在函数 $f\colon X(1)\to \{0,1\}$，使得对每个 $k$ 面 $\sigma$ 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。为简化概率可验证定理（PCP定理）的组件，Goldreich 和 Safra 提出了直接积测试问题：能否仅通过查询 $F$ 的 $2$ 个输入来检验 $F\colon X(k)\to \{0,1\}^k$ 是否与直接积函数相关？Dinur 和 Kaufman 猜想存在有界度复形，其在小可靠性情形下可通过直接积测试。我们通过证明对任意 $\delta>0$，存在一族度为 $O_{\delta}(1)$ 的高维扩展器及一个可靠性为 $\delta$ 的 $2$ 查询直接积测试器，从而解决了该猜想。我们使用了部分作者以及 Dikstein 和 Dinur 独立给出的刻画，该刻画表明某种形式的非阿贝尔上同调边缘扩张（称为"唯一博弈上同调边缘扩张"）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一种处理非阿贝尔群系数复形上同调边缘扩张的通用技术，从而证明 Chapman 和 Lubotzky 构造的高维扩展器满足必要条件，并因此允许具有小可靠性的 $2$ 查询直接积测试器。