Quasipolynomial density bounds for $K$-point configurations in $\mathbb{Z}^d$

Let $d,K,N\in \mathbb{N}$ with $K\geq 3$ and $d\geq 4K+4$. Let $Δ\subset \mathbb{Z}^d$ be the vertex set of a nondegenerate $(K-1)$-simplex, and let $A\subseteq[N]^d$ contain no nontrivial similar copy of $Δ$. We prove that \[ |A|\ll_{Δ,d} N^d\exp\!\left(-c_{Δ,d}\sqrt{\log N}\right) \] impro