NP-Hard Sets Are Exponentially Dense Unless coNP C NP/poly

We show that hard sets S for NP must have exponential density, i.e. |S
_=n
| ges 2n
^epsi
for some isin > 0 and infinitely many n, unless coNP sube NP/poly and the polynomial-time hierarchy collapses. This result holds for Turing reductions that make n
^1-isin
queries. In addition we study the instance complexity o/NP- hard problems and show that hard sets also have an exponential amount of instances that have instance complexity n for some sigma > 0. This result also holds for Turing reductions that make n
^1-isin
queries.