Covering and tiling hypergraphs with tight cycles

Given 3≤k≤s, we say that a k-uniform hypergraph C_s^k is a tight cycle on s vertices if there is a cyclic ordering of the vertices of C_s^k such that every k consecutive vertices under this ordering form an edge. We prove that if k≥3 and s≥2k², then every k-uniform hypergraph on n vertices with minimum codegree at least (1/2+o(1))n has the property that every vertex is covered by a copy of C_s^k. Our result is asymptotically best possible for infinitely many pairs of s and k, e.g. when s and k are coprime. A perfect C_s^k-tiling is a spanning collection of vertex-disjoint copies of C_s^k. When s is divisible by k, the problem of determining the minimum codegree that guarantees a perfect C_s^k-tiling was solved by a result of Mycroft. We prove that if k≥3 and s≥5k² is not divisible by k and s divides n, then every k-uniform hypergraph on n vertices with minimum codegree at least (1/2+1/(2s)+o(1))n has a perfect C_s^k-tiling. Again our result is asymptotically best possible for infinitely many pairs of s and k, e.g. when s and k are coprime with k even.