We prove that for any 3-player game $\mathcal G$, whose query distribution has the same support as the GHZ game (i.e., all $x,y,z\in \{0,1\}$ satisfying $x+y+z=0\pmod{2}$), the value of the $n$-fold parallel repetition of $\mathcal G$ decays exponentially fast: \[ \text{val}(\mathcal G^{\otimes n}) \leq \exp(-n^c)\] for all sufficiently large $n$, where $c>0$ is an absolute constant. We also prove a concentration bound for the parallel repetition of the GHZ game: For any constant $ε>0$, the probability that the players win at least a $\left(\frac{3}{4}+ε\right)$ fraction of the $n$ coordinates is at most $\exp(-n^c)$, where $c=c(ε)>0$ is a constant. In both settings, our work exponentially improves upon the previous best known bounds which were only polynomially small, i.e., of the order $n^{-Ω(1)}$. Our key technical tool is the notion of \emph{algebraic spreadness} adapted from the breakthrough work of Kelley and Meka (FOCS '23) on sets free of 3-term progressions.

翻译：我们证明，对于任意三玩家博弈$\mathcal G$，若其查询分布与GHZ博弈具有相同支撑集（即所有满足$x+y+z=0\pmod{2}$的$x,y,z\in \{0,1\}$），则$\mathcal G$的$n$重并行重复值呈指数级衰减：对于所有充分大的$n$，有\[ \text{val}(\mathcal G^{\otimes n}) \leq \exp(-n^c)\]，其中$c>0$为绝对常数。我们还证明了GHZ博弈并行重复的集中界：对于任意常数$ε>0$，玩家在至少$\left(\frac{3}{4}+ε\right)$比例的$n$个坐标上获胜的概率至多为$\exp(-n^c)$，其中$c=c(ε)>0$为常数。在这两种设定下，我们的工作将先前已知的最佳界限（仅为多项式小量级$n^{-Ω(1)}$）实现了指数级改进。我们的核心技术工具是源自Kelley和Meka（FOCS '23）关于无三项等差数列集合的突破性工作中所采用的\emph{代数扩展性}概念。