Subspaces of ℓ2(X)ℓ2(X) without the approximation property

For a Banach space XX, let {dn(X)}{dn(X)} be the sequence of distances to a Hilbert space. We identify a rather large class of Banach spaces XX with the property that when the rate of growth of dn(X)dn(X) is at least the same as (log(n))β(log(n))β, for some β>1β>1, then XX does not have the hereditary approximation property.