2 problems

Gowers's conjecture. If $\varepsilon$ is sufficiently small in terms of $\delta$ and $c_0$, then there exists a quadratic variety $Q$ such that

Winning conjecture. For every $m\in\mathbb{Q}$, the set $BA_{\partial X}^{\psi}(X_m(\mathbb{Z}))$ is a (strong) winning subset of $\partial X$.