Inducibility in the hypercube

Let Qd ${Q}_{d}$ be the hypercube of dimension d $d$ and let H $H$ and K $K$ be subsets of the vertex set V(Qd) $V({Q}_{d})$, called configurations in Qd ${Q}_{d}$. We say that K $K$ is an exact copy of H $H$ if there is an automorphism of Qd ${Q}_{d}$ which sends H $H$ onto K $K$. Let n >= d $n\ge d$ be an integer, let H $H$ be a configuration in Qd ${Q}_{d}$ and let S $S$ be a configuration in Qn ${Q}_{n}$. We let lambda(H,d,n) $\lambda (H,d,n)$ be the maximum, over all configurations S $S$ in Qn ${Q}_{n}$, of the fraction of sub-d $d$-cubes R $R$ of Qn ${Q}_{n}$ in which S boolean AND R $S\cap R$ is an exact copy of H $H$, and we define the d $d$-cube density lambda(H,d) $\lambda (H,d)$ of H $H$ to be the limit as n $n$ goes to infinity of lambda(H,d,n) $\lambda (H,d,n)$. We determine lambda(H,d) $\lambda (H,d)$ for several configurations in Q3 ${Q}_{3}$ and Q4 ${Q}_{4}$ as well as for an infinite family of configurations. There are strong connections with the inducibility of graphs.