Subspaces of without the approximation property

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