The failure of diamond on a reflecting stationary set

1. It is shown that the failure of {lozenge, open}S, for a set S ⊆ א _ω+1 that reflects stationarily often, is consistent with GCH and APא _ω, relative to the existence of a supercompact cardinal. By a theorem of Shelah, GCH and □* λ entails {lozenge, open}S for any S ⊆ λ+ that reflects stationarily often. 2. We establish the consistency of existence of a stationary subset of [א _ω+1] ^ω that cannot be thinned out to a stationary set on which the supfunction is injective. This answers a question of König, Larson and Yoshinobu in the negative. 3. We prove that the failure of a diamond-like principle introduced by Džamonja and Shelah is equivalent to the failure of Shelah's strong hypothesis.