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Abstract:
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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
probability measure induced by S =
∞n
=1 3−nXn. 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 = 3, E = [2/3, 1], see [6]. It was hoped that this result holds true with
a = 3k, for any k ∈ N. We prove that it is not the case. In fact, our result shows
that for k = 2(a = 6), α = 1, α = 1 − log(1+√5)−log 2
2 log 3 ≈ 0.78099 and E = [1 −
log(1+√5)−log 2
2 log 3 , 1].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
probability measure induced by S =
∞n
=1 3−nXn. 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 = 3, E = [2/3, 1], see [6]. It was hoped that this result holds true with
a = 3k, for any k ∈ N. We prove that it is not the case. In fact, our result shows
that for k = 2(a = 6), α = 1, α = 1 − log(1+√5)−log 2
2 log 3 ≈ 0.78099 and E = [1 −
log(1+√5)−log 2
2 log 3 , 1]. |