Sparse Hard Sets for P: Resolution of a Conjecture of Hartmanis

Building on a recent breakthrough by Ogihara, we resolve a conjecture made by Hartmanis in 1978 regarding the (non-) existence of sparse sets complete for P under logspace many-one reductions. We show that if there exists a sparse hard set for P under logspace many-one reductions, then P = LOGSPACE. We further prove that if P has a sparse hard set under many-one reductions computable in NC1, then P collapses to NC1.