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 comp
lexity of sets A in EXPSPACE that honestly reduce to EXPSPACE-avoiding
sets, we show that if the reducibility used has a property, called ra
nge-constructibility, then A must also reduce to a sparse set under th
e same reducibility.