Fractional edge-decompositions of dense graphs and related eigenvalues

We consider the problem of decomposing some $t$-uniform hypergraph $G$ into copies of another, say $H$, with nonnegative rational weights. For fixed $H$ on $k$ vertices, we show that this is always possible for all $G$ having sufficiently many vertices and `local density' at least $1-C(t)k^{-2t}$. In the case $t=2$ and $H=K_3$, we show that all large graphs with density at least $27/28$ admit a fractional triangle decomposition. The proof relies on estimates of certain eigenvalues in the Johnson scheme.