Hard sets are hard to find

We investigate the frequency of complete sets for various complexity classes within EXP under several polynomial-time reductions in the sense of resource bounded measure. We show that these sets are scarce: The sets that are complete under ⩽(n^α-tt ^-)^P reductions for NP, the levels of the polynomial-time hierarchy, and PSPACE have p₂-measure zero for any constant α<1. The ⩽(n^c-T)^P-complete sets for EXP have p₂ -measure zero for any constant c. Assuming MA≠EXP, the ⩽ _tt^P-complete sets for EXP have p-measure zero. A key ingredient is the Small Span Theorem, which states that for any set A in EXP at least one of its lower span (i.e., the sets that reduce to A) or its upper span (i.e., the sets that A reduces to) has p₂-measure zero. Previous to our work, the theorem was only known to hold for ⩽_btt^p-reductions. We establish it for ⩽(n⁰(1)-tt)^p-reductions