The sets of divergence points of self-similar measures are residual

Let μ be a self-similar measure supported on a self-similar set K with the open set condition. For x ∈ K , let A ( D ( x ) ) be the set of accumulation points of D r ( x ) = log μ ( B ( x , r ) ) log r as r ↘ 0 . In this paper, we show that for any closed non-singleton subinterval I ⊂ R , the set of points x for which the set A ( D ( x ) ) equals I is either empty or residual.