Supercompactness and partial level by level equivalence between strong compactness and strongness
Arthur W. Apter (2004)
Fundamenta Mathematicae
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We force and construct a model containing supercompact cardinals in which, for any measurable cardinal δ and any ordinal α below the least beth fixed point above δ, if is regular, δ is strongly compact iff δ is δ + α + 1 strong, except possibly if δ is a limit of cardinals γ which are strongly compact. The choice of the least beth fixed point above δ as our bound on α is arbitrary, and other bounds are possible.