Christina Giannitsi

Sharp Thresholds for Distance Patterns in Random Subsets of $\mathbb Z^d$

Let $d \geq 5$, $0<\gamma<d - 2$, and $\Omega_N$ be the binomial random subset of $Q_N = [-N,N]^d \cap \mathbb Z^d$ with retention probability $p_N = N^{-\gamma}$. We prove that, with failure probability of optimal exponential order, every subset $B \subseteq \Omega_N$ of fixed positive relative density realizes, at ea...