Separating NE from some nonuniform nondeterministic complexity classes

We investigate the question whether NE can be separated from the reduction closures of tally sets, sparse sets and NP. We show that (1) NE not subset of R-n0(1)-T(NP) (TALLY); (2) NE not subset of R-m(SN) (SPARSE); (3) NEXP not subset of P-nk-T(NK)/n(k) for all k >= 1; and (4) NE not subset of P-btt (NP circle plus SPARSE). Result (3) extends a previous result by Mocas to nonuniform reductions. We also investigate how different an NE- hard set is from an NP- set. We show that for any NP subset A of a many- one- hard set H for NE, there exists another NP subset A' of H such that A' superset of A and A' - A is not of sub-exponential density.