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

Abstract

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 \[ |A|\ll_{Δ,d} N^d\exp\!\left(-c_{Δ,d}\sqrt{\log N}\right) \] improving upon a polylogarithmic bound due to Magyar. We perform a density increment argument using the circle method, and we introduce a ``cut operator'' method to decouple the weighted exponential sum over the system of quadratic forms describing the simplex. Our proof combines ideas from graph theory, functional analysis, and the geometry of numbers. In the process, we apply Finner's fractional form of Hölder's inequality, the analytic large sieve, and Kim's mean value formula for primitive lattice flags.