Page 66 - Textos de Matemática Vol. 34

56 SUSANA D. MOURA
always to mean that there are two positive numbers c1 and c2 such that c1ak ≤bk ≤c2ak or c1φ(x)≤ψ(x)≤c2φ(x)
for all admitted values of the discrete variable k or the continuous variable x, where (ak)k, (bk)k are non-negative sequences and φ, ψ are non-negative functions. Occasionally, we shall also consider the stronger equivalence “≈” in ak ≈ bk or φ(x) ≈ ψ(x) to mean that
ak→1 or φ(x)→1, bk ψ(x)
as the discrete variable k tends to infinity, or the continuous variable x tends to zero or to infinity (depending on the context), respectively. All unimportant positive constants will be denoted by c, occasionally with additional subscripts within the same formula. If not otherwise indicated, log is always taken with respect to base 2.
2. (d,Ψ)-sets
In this section we specify the class of fractals we shall take into consideration.
We begin by recalling the concept of slowly varying functions.
Definition 2.1. A positive and measurable function Ψ defined on the interval (0, 1]
is said to be slowly varying if
lim Ψ(sr) = 1, s ∈ (0,1].
r→0 Ψ(r)
Examples 2.1. Any function of the form 1 ds
Ψ(r)=exp − ε(s) s , r∈(0,1], r
where ε is a measurable function with lims→0 ε(s) = 0, is slowly varying (actually this is a characterisation: any slowly varying function is equivalent to a function Ψ of the above type for an appropriate function ε); in particular,
Ψk,b(r) = logk(cr−1)b , r ∈ (0, 1],
where b ∈ R, logk denotes the k-times iterated logarithm, and c is a large enough constant, is slowly varying. We remark that Ψk,b is also an example of an admissible function in the sense of [16, 17]. We recall that an admissible function Ψ is a positive monotone function defined on (0,1] such that Ψ(2−2j) ∼ Ψ(2−j), j ∈ N0. An admissible function is, up to equivalence, a slowly varying function (cf. Proposition 1.9.7 of [4]).