Let μ be the probability measure induced by S =
∞n
=1 3−nXn, where
X1,X2, ... is a sequence of independent, identically distributed (i.i.d) random variables
each taking values 0, 1, a with equal probability 1/3. Let α(s) (resp.α(s), α(s)) denote
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};
E = {α : α(s) = α for some s ∈ supp μ}
.
In the case a = 3k +1 for k = 1, E = [1 − log(1+√5)−log 2
log 3 , 1], see [10]. It is conjectured
that in the general case, for a = 3k + 1 ( k ∈ N), the local dimension is of the form as
the case k = 1, i.e., E = [1 − log a
b log 3 , 1] for a, b depends on k. In fact, our result shows
that for k = 2 (a = 7), we have α = 1, α = 1− log(1+√3)
3 log 3 and E = [1 − log(1+√3)
3 log 3 , 1].
