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Let X1 , X2 , . be a sequence of independent, identically distributed(i.i.d) random variables each taking values 0, 1, a with equal probability 1/3. Let µ be the ∞ −n probability measure induced by S = Xn . Let α(s) (resp.α(s), α(s)) denote n=1 3 the local dimension (resp. lower, upper local dimension) of s ∈ supp µ, and let α = sup{α(s) : s ∈ supp µ}; α = inf{α(s) : s ∈ supp µ} | VNU. JOURNAL OF SCIENCE Mathematics - Physics. T.XXI N01 - 2005 LOCAL DIMENSION OF FRACTAL MEASURE ASSOCIATED WITH THE 0 1 a - PROBLEM THE CASE a 6 Le Xuan Son Pham Quang Trinh Vinh University Nghe An Vu Hong Thanh Pedagogical College of Nghe An Abstract. Let X1 X2 . be a sequence of independent identically distributed i.i.d random variables each taking values 0 1 a with equal probability 1 3. Let g be the probability measure induced by S 2m i 3-nXn. Let a s resp.a s a s denote the local dimension resp. lower upper local dimension of s E supp g and let a sup a s s E supp g a inf a s s E supp g E a a s a for some s E supp g . In the case a 3 E 2 3 1 see 6 . It was hoped that this result holds true with a 3k for any k E N. We prove that it is not the case. In fact our result shows that for k 2 a 6 a 1 a 1 - log 1 2 og 3 log 2 0.78099 and E 1 -log 1 V5 -log2 1 2 log 3 1 . 1. Introduction Let X1 X2 . be a sequence of i.i.d random variables each taking values a1 a2 . am with probability P1 P2 . pm respectively. Then the sum S 2 pnXn n 1 is well defined for 0 p 1. Let g be the probability measure induced by S i.e. g A Prob w S w E A . It is known that the measure g is either purely singular or absolutely continuous. In 1996 Lagarias and Wang 8 showed that if m is a prime number p1 P2 . pm 1 m and a1 . am are integers then g is absolutely if and only if a1 a2 . am forms a complete system modm i.e. a1 0 mod m a2 1 mod m . am m 1 mod m . An intriguing case when m 3 p1 P2 P3 3 and a1 0 a2 1 a3 3 known as the 0 1 3 Problem is of great interest and has been investigated since the last decade. Typeset by AmS-Ti X 31 32 Le Xuan Son Pham Quang Trinh Vu Hong Thanh Let us recall that for s E supp ự the local dimention a s of ự at s is defined by X _ logự Bh s m a s lim ------ 7----- 1 h A0 log h provided that 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 denoted a s and a s by .
