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

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