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, α, ε)-pseudo-random if for all not necessarily disjoint sets A₁, . . ., A_k ⊂ V (H) with ∣A₁∣∙∙∙∣A_k∣ > αn^k we have e(A₁, . . ., A_k) = (1 ± ε)p∣A₁∣∙∙∙∣A_k∣. For any linear k-uniform F we provide a bound on α = α(n) in terms of p = p(n) and F, such that (under natural divisibility assumptions on n) any (p, α, o(1))-pseudorandom n-vertex H with a mild minimum 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 of our results, perfect matchings appear at α = o(p^k) while loose Hamilton cycles appear at α = o(p^k−1). This extends the works of Lenz-Mubayi, and Lenz-Mubayi-Mycroft who studied the analogous problems in the dense setting.