On Reductions to Sets that Avoid EXPSPACE

A set B is called EXPSPACE-avoiding, if every subset of B in EXPSPACE is sparse. For example, sets of high information density (called HIGH sets in 5]) are shown to be EXPSPACE-avoiding. Investigating the complexity of sets A in EXPSPACE that honestly reduce to EXPSPACE-avoiding sets, we show that if the reducibility used has a property, called range-constructibility, then A must also reduce to a sparse set under the same reducibility.