Let λ denote a singular cardinal. Zeman, improving a previous result of Shelah, proved that □*λ together with 2λ=λ⁺ implies ♢S for every S⊆λ⁺ that reflects stationarily often. In this paper, for a set S⊆λ⁺, a normal subideal of the weak approachability ideal is introduced, and denoted by I[S;λ]. We say that the ideal is fat if it contains a stationary set. It is proved:
1. if I[S;λ] is fat, then NSλ⁺↾ S is non-saturated;
2. if I[S;λ] is fat and 2λ=λ⁺, then ♢S holds;
3. □*λ implies that I[S;λ] is fat for every S⊆λ⁺ that reflects stationarily often;
4. it is relatively consistent with the existence of a supercompact cardinal that □*λ fails, while I[S;λ] is fat for every stationary S⊆λ⁺ that reflects stationarily often.
The stronger principle ♢*λ⁺ is studied as well.
"A relative of the approachability ideal, diamond and non-saturation." J. Symbolic Logic 75 (3) 1035 - 1065, September 2010. https://doi.org/10.2178/jsl/1278682214