Improved Parallel Repetition for GHZ-Supported Games via Spreadness

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 $x+y+z=0(\mathrm{m}\mathrm{o}\mathrm{d}2)$), the value of the $n$-fold parallel repetition of $\mathcal{G}$ decays exponentially fast: $$\text{val}({\mathcal{G}}^{\otimes n})\le \exp (-n^c)$$ 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 $\epsilon >0$, the probability that the players win at least a $(\frac{3}{4}+\epsilon )$ fraction of the $n$ coordinates is at most $\exp (-n^c)$, where $c=c(\epsilon )>0$ 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 $n^{-\mathrm{\Omega }(1)}$. 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.