恒定度直接积测试器与小健全性
Constant Degree Direct Product Testers with Small Soundness
AI总结：
该论文解决了 Dinur 和 Kaufman 的猜想，通过证明存在恒定度高维扩展器族支持小健全性的 2 查询直接积测试器，并提出了非阿贝尔上同调边界扩张的通用证明技术。
摘要：
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 PCP 定理的组成部分，Goldreich 和 Safra 引入了直接积测试问题，该问题询问是否可以通过仅查询 $F$ 的 $2$ 个输入来测试 $F\colon X(k)\to \{0,1\}^k$ 是否与直接积函数相关。Dinur 和 Kaufman 猜想存在有界度复形，其在小健全性机制下具有直接积测试。我们通过证明对于所有 $\delta>0$，存在一族度为 $O_\delta(1)$ 的高维扩展器和一个健全性为 $\delta$ 的 $2$ 查询直接积测试器来解决他们的猜想。我们使用了部分作者及独立地 Dikstein 和 Dinur 给出的刻画，他们证明了某种形式的非阿贝尔上同调边界扩张（他们称之为“Unique-Games 上同调边界扩张”）是复形允许此类直接积测试器的必要且充分条件。我们的主要技术贡献是一种用于证明具有非阿贝尔群系数的复形上同调边界扩张的通用技术。这使我们能够证明 Chapman 和 Lubotzky 构造的高维扩展器满足必要条件，从而允许具有小健全性的 $2$ 查询直接积测试器。
英文摘要：
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 $F(σ) = (f(σ_1), \ldots, f(σ_k))$ for each $k$-face $σ$. 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 $δ>0$, there exists a family of high-dimensional expanders with degree $O_δ(1)$ and a $2$-query direct product tester with soundness $δ$. 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.
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