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Acta Mathematica Vietnamica

THE EXTREME VALUES OF LOCAL DIMENSION IN FRACTAL GEOMETRY

icon-email VU THI HONG THANH

Abstract

Let μ be the probability measure induced by S=i=0qiXi, where mq2 are integers and X0,X1, a sequence of independent identically distributed random variables each taking integer values 0,1,,m with equal probability p=1/(m+1). Let α(s) (resp. α(s),α(s)) denote the local dimension (resp. lower, upper local dimension) of ssuppμ, and let α=sup{α(s):ssuppμ};α=inf{α(s):ssuppμ}. We show that α=log(m+1)log(r+r2+4(l+1))+log2logq for rqm<rq+r; r=1,,q1 and l=mrq.

The special case of our result, m=rq (l=0), was obtained earlier in [6].