Research proof index

Fix a frequency parameter $\vartheta$, a level $N$, and a spatial grid cube $L$. Among the auxiliary tiles attached to an antichain $\mathfrak{A}$ whose spatial cube is exactly $L$, the total measure of their active sets inside $G$ is at most

$2^{a(N+5)}\,\mathrm{dens}_1(\mathfrak{A})\,\mu(L).$