Let X be random variable taking values 0, 1, a with equal probability 1/3
and let X1,X2, ... be a sequence of independent identically distributed (i.i.d) random
variables with the same distribution as X. Let μ be the probability measure induced by
S =
∞i
=1 3−iXi. Let α(s) (resp. α(s), α(s)) denote the local dimension (resp. lower,
upper local dimension) of s ∈ supp μ.
Put
α = sup{α(s) : s ∈ supp μ}; α = inf{α(s) : s ∈ supp μ};
E = {α : α(s) = α for some s ∈ supp μ}.
When a ≡ 0 (mod 3), the probability measure μ is singular and it is conjectured that for
a = 3k (for any k ∈ N), the local dimension is still the same as the case k = 1, 2. Itmeans
E = [1− log(a)
b log 3 , 1], for a, b depend on k. Our result shows that for k = 3 (a = 9), α = 1,
α = 2/3 and E = [2
3 , 1]Let X be random variable taking values 0, 1, a with equal probability 1/3
and let X1,X2, ... be a sequence of independent identically distributed (i.i.d) random
variables with the same distribution as X. Let μ be the probability measure induced by
S =
∞i
=1 3−iXi. Let α(s) (resp. α(s), α(s)) denote the local dimension (resp. lower,
upper local dimension) of s ∈ supp μ.
Put
α = sup{α(s) : s ∈ supp μ}; α = inf{α(s) : s ∈ supp μ};
E = {α : α(s) = α for some s ∈ supp μ}.
When a ≡ 0 (mod 3), the probability measure μ is singular and it is conjectured that for
a = 3k (for any k ∈ N), the local dimension is still the same as the case k = 1, 2. Itmeans
E = [1− log(a)
b log 3 , 1], for a, b depend on k. Our result shows that for k = 3 (a = 9), α = 1,
α = 2/3 and E = [2
3 , 1]
