Factors and loose Hamilton cycles in sparse pseudo-random hypergraphs

We investigate the emergence of spanning structures in sparse pseudo-random $k$-uniform hypergraphs, using the following comparatively weak notion of pseudo-randomness. A $k$-uniform hypergraph $H$ on $n$ vertices is called $(p,\alpha ,ϵ)$-pseudo-random if for all (not necessarily disjoint) vertex subsets $A_1,\dots ,A_k\subseteq V(H)$ with $|A_1|\cdots |A_k|{\geq}\alpha n^{k}$wehave$$e(A_1,\dots, A_k)=(1\pm\epsilon)p |A_1|\cdots |A_k|.$$Foranylinear$k$-uniform$F$weprovideaboundon$\alpha=\alpha(n)$intermsof$p=p(n)$and$F$, such that (under natural divisibility assumptions on $n$) any $k$-uniform $\big(p,\alpha, o(1)\big)$-pseudo-random$n$-vertexhypergraph$H$ with a mild minimum vertex degree condition contains an $F$-factor. The approach also enables us to establish the existence of loose Hamilton cycles in sufficiently pseudo-random hypergraphs and all results imply corresponding bounds for stronger notions of hypergraph pseudo-randomness such as jumbledness or large spectral gap. As a consequence, $(p,\alpha ,o(1))$-pseudo-random $k$-graphs as above contain: $(i)$ a perfect matching if $\alpha =o(p^k)$ and $(ii)$ a loose Hamilton cycle if $\alpha =o(p^{k-1})$. This extends the works of Lenz–Mubayi, and Lenz–Mubayi–Mycroft who studied the analogous problems in the dense setting.