... the limit exists, where Bh(s) denotes the ball centered at s with radius h.If the limit (1) does not exist, we define the upper and lower local dimension, denotedα(s)and α(s), by taking the ... In Section 2 we establish some auxiliary results used in the proof of the Main Theorem. The proof o f the Main Theorem will be given in the lastsection.2. Some Auxiliary ResultsLet X1,X2, ... Ngai, Iterated function systems with overlaps, AsianJ. Math. 4(2000), 527 - 552.4. T. Hu, The local dimensions of the Bernoull i convolution associated with the goldennumber, Trans. Amer....