SOME REMARKS ON ENTROPY NUMBERS OF SETS WITH FRACTIONAL DIMENSION

GIORGIO FOLLO

Abstract

Entropy numbers are frequently used to study certain compactness properties of linear and continuous operators between Banach spaces.

Entropy numbers are first defined for bounded subsets of metric spaces. Then the definition is extended to linear and continuous operators by considering the image of the closed unit ball. We study entropy numbers of sets and generalize some results shown by B. Carl and I. Stephani in [1].

It is well known that if Ω is a unit closed ball in the Euclidean space R^{N} , then for every n ∈ N ,

n^{− \frac{1}{N}} ≤ ε_{n}(Ω) ≤ 4n^{− \frac{1}{N}} .

We extend estimates of this type to the following cases:

• Ω is a closed ball in a space of homogeneous type or in a β-space.

• Ω is a fractal generated by a finite system of contractions in a complete metric space.

• Ω is a bounded d-set in R^{N} . We recall that every self-similar fractal set is a d-set.

Moreover we study relations between the asymptotic behaviour of the entropy numbers of bounded subsets of R^{N} and their box dimension.

A straightforward consequence is that the Hausdorff dimension and the box dimension of a bounded d-set in R^{N} coincide.

1. Introduction

Let (X, D) be a metric space and let Ω be a bounded subset of X . The n-th entropy number of Ω is defined by

ε_{n}(Ω) = inf{ε > 0 | ∃ x_{1}, x_{2}, ..., x_{n} ∈ X such that Ω ⊆ ⋃_{i=1}^{n} B_{X}(x_{i}, ε)} ,

where B_{X}(x_{i}, ε) = {x ∈ X | D(x, x_{i}) ≤ ε} denotes the closed ball centered in x_{i} and of radius ε ≥ 0 . The sequence (ε_{n}(Ω)) is monotonically decreasing and tends to zero if and only if Ω is precompact.

The numbers ε_{n}(Ω) are also called the outer entropy numbers of Ω ; inner entropy numbers are also useful and they are defined by

φ_{n}(Ω) = sup{ρ ≥ 0 | ∃ y_{1}, y_{2}, ..., y_{n+1} ∈ Ω such that
D(y_{i}, y_{j}) > 2ρ ∀i, j ∈ N with i < j ≤ n + 1} .

1991 Mathematics Subject Classification. Primary: 28A80; Secondary: 28A78. Key words and phrases. Fractals.

If ♯Ω ≤ n we put φ_{n}(Ω) = 0 .

The following relationship between inner and outer entropy numbers will be frequently used in this paper:

φ_{n}(Ω) ≤ ε_{n}(Ω) ≤ 2φ_{n}(Ω) ∀n ∈ N . (1)

See [1], chapter 1, for a complete treatment.

We will study the connections between the asymptotic behaviour of the entropy numbers of bounded subsets of R^{N} and their box and Hausdorff dimension. We will show that if Γ ⊆ R^{N} is a bounded d-set, then there exist a, b > 0 such that

an^{− \frac{1}{d}} ≤ ε_{n}(Γ) ≤ bn^{− \frac{1}{d}}

for every n ∈ N . If the above estimate is valid for the entropy numbers of a bounded subset of R^{N} (not necessarily a d-set) , then its box counting dimension is d .

We will also prove upper estimates of the same type for sets that are generated by a finite system of bi-Lipschitz contractions.

An upper estimate holds in more general cases, while our proof of a lower estimate requires more restrictive hypotheses.

Finally, in sections 4 and 5, we will give upper estimates for closed balls in β-spaces and spaces of homogeneous type.

This research was supported by M.U.R.S.T. and by Universities of Milano and Pavia.

I wish to thank I. Stephani and C. Richter, who invited me to give a lecture on these topics in Jena, C. Bondioli and M. P. Bernardi, for helpful discussion on this subject.

2. Definitions and preliminary results

Throughout this paper (X, D) denotes a complete separable metric space.

If X = R^{N} then we will use the Euclidean metric: D(x, y) =

||x − y||_{2} = \sqrt{∑_{i=1}^{N}(ξ_{i} − η_{i})^{2}} , where x = (ξ_{1}, ξ_{2}, ..., ξ_{N}) and y = (η_{1}, η_{2}, ..., η_{N}) .

N stands for the set of integer numbers strictly greater than 0 . For a map f : X → X the terms Lipschitz continuous function,

contraction, similitude have the usual meaning.

If A is a subset of X , we denote the diameter of A by |A| and the number of elements of A by ♯A . If s ≥ 0 , ℋ^{s}(A) stands for the s-dimensional Hausdorff measure of A and dimA for its Hausdorff dimension.

2.1. Contractions and fractals

Let 𝔉 = {f_{1}, f_{2}, ..., f_{M}} be a finite family of contraction in (X, D) .

We set ρ_{i} = inf{c > 0 | D(f_{i}(x), f_{i}(y)) ≤ c D(x, y) ∀x, y ∈ X} (ρ_{i} is the Lipschitz constant of f_{i} ), ρ = max{ρ_{1}, ρ_{2}, ..., ρ_{M}} , σ_{i} = sup{c ≥ 0 | D(f_{i}(x), f_{i}(y)) ≥ c D(x, y) ∀x, y ∈ X} , σ = min{σ_{1}, σ_{2}, ..., σ_{M}} .

Moreover, we will simply denote the contraction f_{i_{1}} ∘ f_{i_{2}} ∘ ... ∘ f_{i_{n}} by f_{i_{1}i_{2}...i_{n}} (where f_{i_{j}} ∈ 𝔉 for 1 ≤ j ≤ n ) and the fixed point of f_{i_{1}i_{2}...i_{n}} by x_{i_{1}i_{2}...i_{n}} .

By induction, it is easy to prove that

(∏_{j=1}^{n} σ_{i_{j}}) D(x, y) ≤ D(f_{i_{1}i_{2}...i_{n}}(x), f_{i_{1}i_{2}...i_{n}}(y)) ≤

≤ (∏_{j=1}^{n} ρ_{i_{j}}) D(x, y) ∀x, y ∈ X (2)

and f_{i_{1}i_{2}...i_{n}} is injective if σ > 0 .

If A is a subset of X we denote by 𝔉(A) the set ⋃_{i=1}^{M} f_{i}(A) . The following theorems, due to J. E. Hutchinson, play a fundamental

role for the results exposed in the next sections.

Theorem 1. There exists a unique non-empty compact set K ⊆ X invariant with respect to 𝔉 , i. e. such that

K = 𝔉(K) . (3)

PROOF. - See [5], chapter 3.

We call K the fractal associated to the system 𝔉 .

If n, k ∈ N with k < n , i_{1}, i_{2}, ..., i_{n} ∈ {1, 2, ..., M} , then (3) shows that f_{i_{1}i_{2}...i_{n}}(K) ⊆ f_{i_{1}i_{2}...i_{k}}(K) ⊆ K .

In [5] is also shown that K is the closure of the set {x_{i_{1}i_{2}...i_{n}} ∈ X | n ∈ N , i_{1}, i_{2}, ..., i_{n} ∈ {1, 2, ..., M}} . Moreover, for each sequence (i_{j}) of elements of {1, 2, ..., M}, the set ⋂_{n=1}^{+∞} f_{i_{1}i_{2}...i_{n}}(K) is a singleton and K is the union of these singletons.

Theorem 2. Let r = {r_{1}, r_{2}, ..., r_{M}} be a family of M real numbers in ]0,1[ with ∑_{i=1}^{M} r_{i} = 1 . Then there exists a unique Borel regular (outer) measure μ in X , compactly supported and of total mass 1 , such that μ is invariant with respect to (𝔉, r) , i. e.

μ(A) = ∑_{i=1}^{M} r_{i}μ(f_{i}^{−1}(A)) ∀A ∈ X . (4)

Furthermore the support of μ is the fractal K . PROOF. - See [5], chapter 4.

We are interested in the following invariant measure: let d_{ρ} > 0 be the unique real number defined by ∑_{i=1}^{M} ρ_{i}^{d_{ρ}} = 1 ; we denote by μ_{𝔉} the invariant measure with respect to (𝔉, {ρ_{1}^{d_{ρ}}, ρ_{2}^{d_{ρ}}, ..., ρ_{M}^{d_{ρ}}}) .

If X = R^{N} with the Euclidean distance and the contraction f_{1}, f_{2}, ..., f_{n} are similitudes, then the number d_{ρ} is called the similarity dimension of 𝔉 .

The system 𝔉 is said to satisfie the Open Set Condition (O.S.C.) if there exists a bounded open set O ⊆ X such that f_{i}(O) ⊆ O f_{i}(O) ⋂ f_{j}(O) = ∅ for i ≠ j (i, j = 1, 2, ..., M).

2.2. d-sets in R^{N}

Definition 3. Let Γ be a closed non-empty subset of R^{N} and let d ∈]0, N]. A positive Borel outer measure μ with support Γ is called a d-measure on Γ if there exist c_{1}, c_{2} > 0 such that for every x_{0} ∈ Γ and for every r ∈]0, 1]

c_{1}r^{d} ≤ μ(B_{R^{N}} (x_{0}, r)) ≤ c_{2}r^{d} ∀x_{0} ∈ Γ ∀r ∈]0, 1] (5)

holds.

Remark 1. One can replace the condition r ∈]0, 1] in the above definition with the condition r ∈]0, r_{0}] ; obviously the constants c_{1} and c_{2} will be replaced by some constants c_{1}(r_{0}) > 0 and c_{2}(r_{0}) > 0 depending on r_{0} .

Remark 2. If μ is a d-measure on Γ , then μ(B_{R^{N}} (x_{0}, r)) ≤ 2^{d}c_{2}r^{d} for every x_{0} ∈ R^{N} and for every r ∈]0, r_{0}/2] .

Definition 4. A closed non-empty subset Γ of R^{N} is called a d-set if there exists a d-measure on Γ .

It can be proved that if Γ is a d-set, μ_{1} and μ_{2} are d-measure on Γ , then there exist constants a, b > 0 such that aμ_{1}(A) ≤ μ_{2}(A) ≤ bμ_{1}(A) ∀A ⊆ R^{N}. Moreover, the restriction to Γ of the Hausdorff measure ℋ^{d} is a d-measure on Γ ; so every d-set has its ”canonical” d-measure and therefore d is unique.

For the proof of these facts see [6], chapter 1.

Remark 3. The following condition (6) is frequently used to describe density properties of a measure μ in R^{N} , with respect to a set Ω ⊆ R^{N} and a real number d ∈]0, N[ :

There are a, b ∈]0, +∞[ such that for every x ∈ Ω

(6) a ≤ lim inf_{r→0^{+}} \frac{μ(B_{R^{N}} (x, r) ⋂ Ω)}{r^{d}} ≤ lim sup_{r→0^{+}} \frac{μ(B_{R^{N}} (x, r) ⋂ Ω)}{r^{d}} ≤ b .

It is clear that if Ω is a d-set for a measure μ , then (6) will be satisfied.

By the following simple example we can show that the condition (6) is weaker than the one of being a d-set, even if Ω is compact.

Example 1. Let N = 2 and A = {(x, y) ∈ R^{2} | − x^{2} ≤ y ≤ x^{2}} ; if μ is the Lebesgue measure on R^{2} and Ω = (A ⋃(] − ∞, 0] × R)) ⋂([−2, 2]^{2}) then

\frac{π}{4} ≤ lim inf_{r→0^{+}} \frac{μ(B_{R^{N}} ((x, y), r) ⋂ Ω)}{r^{2}} ≤ lim sup_{r→0^{+}} \frac{μ(B_{R^{N}} ((x, y), r) ⋂ Ω)}{r^{2}} ≤ π

∀(x, y) ∈ Ω ;

but Ω is not a 2-set.

In fact, let us suppose that μ|_{Ω} is a 2-measure for Ω ; then there exists a ∈]0, \frac{π}{4}[ such that

∀(x, y) ∈ Ω , ∀r ∈]0, 1[ \frac{μ(B_{R^{N}} ((x, y), r) ⋂ Ω)}{r^{2}} ≥ a ;

but if we set x = r = a/16 and y = 0 , then it will follow

\frac{μ(B_{R^{N}} ((x, y), r) ⋂ Ω)}{r^{2}} = \frac{256}{a^{2}}μ (B ((\frac{a}{16}, 0) , \frac{a}{16}) ⋂ Ω) <

< \frac{512}{a^{2}} ∫_{0}^{a/8} x^{2}dx = \frac{a}{3} < a .

Since μ is equivalent to the measure ℋ^{2} , it follows that Ω is not a 2-set.

Remark 4. We recall that every bounded d-set in R^{N} is Ahlfors regular of dimension d (see [3], pag. 1 for the definition of Ahlfors regular metric space). However these two notions are different (for example, the 2-set [0, 1] × R ⊆ R^{2} is not Ahlfors regular).

2.3. Contractive similitudes.

If X = R^{N} , D is the Euclidean distance and the contraction maps are also similitudes, we can use an additional result about K and μ_{𝔉} .

Theorem 5. Suppose that f_{1}, f_{2}, ..., f_{M} are similitudes and that the system 𝔉 satisfies the open set condition.

Then K is a d_{ρ}-set and μ_{𝔉} = (ℋ^{d_{ρ}}(K))^{−1}ℋ^{d_{ρ}}|_{K} .

More precisely, let O be an open set verifying the open set condition. Suppose that O contains a ball of radius a_{1} and is contained in a ball of radius a_{2}. Then for every x_{0} ∈ K and r ∈]0, |K|] , we have

(\frac{σ}{|K|})^{d_{ρ}} r^{d_{ρ}} ≤ μ_{𝔉}(B_{R^{N}} (x_{0}, r)) ≤ (\frac{1 + 2a_{2}}{σa_{1}} )^{N} r^{d_{ρ}} . (7)

PROOF. - See [5], chapter 5.

2.4. Box-counting dimensions

Here we recall the definition and some characterizations of the boxcounting dimension of a non-empty bounded subset of R^{N} . This notion is closely related to entropy numbers.

In this section Ω will be a non-empty bounded subset of R^{N} .

Definition 6. For all δ > 0 define

N_{δ}(Ω) = min{n ∈ N | ∃C_{1}, C_{2}, ..., C_{n} ⊆ R^{N} so that

Ω ⊆ ⋃_{i=1}^{n} C_{i} and |C_{i}| ≤ δ ∀i ∈ N i ≤ n} .

The lower and upper box-counting dimensions of Ω are defined as

\underline{dim_{B}}Ω = lim inf_{δ→0^{+}} \frac{ln N_{δ}(Ω)}{− ln δ} ,

\overline{dim_{B}}Ω = lim sup_{δ→0^{+}} \frac{ln N_{δ}(Ω)}{− ln δ} .

If \underline{dim_{B}}Ω = \overline{dim_{B}}Ω , their common value is called the box-counting dimension of Ω and is denoted by dim_{B}Ω .

Definition 7. For all δ > 0 define

N(Ω, δ) = min{n ∈ N | ∃x_{1}, x_{2}, ..., x_{n} ∈ R^{N} so that Ω ⊆ ⋃_{i=1}^{n} B_{R^{N}} (x_{i}, δ)} ,

M(Ω, δ) = max{n ∈ N | ∃y_{1}, y_{2}, ..., y_{n} ∈ Ω so that
||y_{i} − y_{j}||_{2} ≥ 2δ ∀i, j ∈ N i < j ≤ n} .

N(Ω, δ) is commonly referred as the entropy function of Ω .

Proposition 8. We have

\underline{dim_{B}}Ω = lim inf_{δ→0^{+}} \frac{ln N(Ω, δ)}{− ln δ} = lim inf_{δ→0^{+}} \frac{ln M(Ω, δ)}{− ln δ} ,

\overline{dim_{B}}Ω = lim sup_{δ→0^{+}} \frac{ln N(Ω, δ)}{− ln δ} = lim sup_{δ→0^{+}} \frac{ln M(Ω, δ)}{− ln δ} .

Moreover, if (δ_{n}) is a sequence of real numbers such that lim_{n} δ_{n} = 0 and there exist a constant c ∈]0, 1[ so that

0 ≤ cδ_{n} ≤ δ_{n+1} ≤ δ_{n} ∀n ∈ N , (8)

then

\underline{dim_{B}}Ω = lim inf_{n} \frac{ln N_{δ_{n}}(Ω)}{− ln δ_{n}} = lim inf_{n} \frac{ln N(Ω, δ_{n})}{− ln δ_{n}} = lim inf_{n} \frac{ln M(Ω, δ_{n})}{− ln δ_{n}} (9)

and

\overline{dim_{B}}Ω = lim sup_{n} \frac{ln N_{δ_{n}}(Ω)}{− ln δ_{n}} = lim sup_{n} \frac{ln N(Ω, δ_{n})}{− ln δ_{n}} =

(10)  lim sup_{n} \frac{ln M(Ω, δ_{n})}{− ln δ_{n}} .

PROOF. - See [4], § 3.1.

We did not find the following results in the literature. Probably some of them are already known, since they are straightforward.

Corollary 9. Let d ∈]0, N] . If there exist b > 0 and c ∈]0, 1[ such that cε_{n}(Ω) ≤ ε_{n+1}(Ω) ≤ b(n + 1)^{− \frac{1}{d}} for all n ∈ N , then \overline{dim_{B}}Ω ≤ d . PROOF. - We suppose ε_{n}(Ω) > 0 ∀n ∈ N ; otherwise ♯Ω < +∞ and \overline{dim_{B}}Ω = 0 < d .

We apply (10) with δ_{n} = \frac{3}{2}ε_{n}(Ω) . The sequence (δ_{n}) satisfies the condition (8); moreover, ∀n ∈ N N(Ω, δ_{n}) ≤ n , because δ_{n} > ε_{n}(Ω) . Then

\frac{ln N(Ω, δ_{n})}{− ln δ_{n}} ≤ \frac{ln n}{− ln \frac{3}{2}ε_{n}(Ω)} ≤ \frac{ln n}{− ln \frac{3}{2}bn^{− \frac{1}{d}}} = \frac{d ln n}{ln n − d ln \frac{3}{2}b}

and

\overline{dim_{B}}Ω ≤ lim sup_{n} \frac{d ln n}{ln n − d ln \frac{3}{2}b} = d ■

Corollary 10. Let d ∈]0, N] . If there exist a > 0 and c ∈]0, 1[ such that acn^{− \frac{1}{d}} ≤ cε_{n}(Ω) ≤ ε_{n+1}(Ω) for all n ∈ N , then \underline{dim_{B}}Ω ≥ d . PROOF. - We apply (9) with δ_{n} = \frac{1}{2}φ_{n}(Ω) . For all n ∈ N we have \frac{c}{2}δ_{n} = \frac{c}{4}φ_{n}(Ω) ≤ \frac{c}{4}ε_{n}(Ω) ≤ \frac{1}{4}ε_{n+1}(Ω) ≤ \frac{1}{2}φ_{n+1}(Ω) = δ_{n+1} , so the sequence (δ_{n}) satisfies the condition (8); moreover for every n ∈ N we have M(Ω, δ_{n}) ≥ n + 1 because δ_{n} < φ_{n}(Ω) ; then

\frac{ln M(Ω, δ_{n})}{− ln δ_{n}} ≥ \frac{ln(n + 1)}{− ln \frac{1}{2}φ_{n}(Ω)} ≥ \frac{ln(n + 1)}{− ln \frac{1}{4}ε_{n}(Ω)} ≥ \frac{ln(n + 1)}{− ln \frac{a}{4}n^{− \frac{1}{d}}} = \frac{d ln(n + 1)}{ln n − d ln \frac{a}{4}}

and

\underline{dim_{B}}Ω ≥ lim inf_{n} \frac{d ln(n + 1)}{ln n − d ln \frac{a}{4}} = d ■

Corollary 11. Let d ∈]0, N] . If there exist a, b ∈]0, +∞[ such that an^{− \frac{1}{d}} ≤ ε_{n}(Ω) ≤ bn^{− \frac{1}{d}} for all n ∈ N , then dim_{B} Ω = d . PROOF. - Let n ∈ N . We have

\frac{ε_{n}(Ω)}{ε_{n+1}(Ω)} ≤ \frac{bn^{− \frac{1}{d}}}{a(n + 1)^{− \frac{1}{d}}} .

Then, for every n ∈ N

\frac{a^{2}}{2^{\frac{1}{d}} b}n^{− \frac{1}{d}} ≤ \frac{a}{2^{\frac{1}{d}} b}ε_{n}(Ω) ≤ \frac{a}{b} (\frac{n}{n + 1})^{\frac{1}{d}} ε_{n}(Ω) ≤ ε_{n+1}(Ω) ≤ b(n + 1)^{− \frac{1}{d}}

and we can apply Corollaries 9 and 10 ■

8 GIORGIO FOLLO

Corollary 12. If \underline{dim_{B}}Ω ≥ d > 0 , then for all ε ∈]0, d[ there exists n_{ε} ∈ N so that ε_{n}(Ω) ≥ n^{− \frac{1}{d−ε}} for n > n_{ε} . PROOF. - Let δ_{n}^{(k)} = (1 + 1/k)ε_{n}(Ω) ∀n, k ∈ N . By Proposition 8

lim inf_{n} \frac{ln N(Ω, δ_{n}^{(k)})}{− ln δ_{n}^{(k)}} ≥ d ∀k ∈ N .

Given ε ∈]0, d[ , there exists n_{ε} ∈ N such that

\frac{ln N(Ω, δ_{n}^{(k)})}{− ln δ_{n}^{(k)}} ≥ d − ε and δ_{n}^{(k)} < 1 ∀n, k ∈ N with n > n_{ε} .

( n_{ε} can be chosen independently from k since δ_{n}^{(k)} > δ_{n}^{(k+1)} ∀n, k ∈ N).

Let n ∈ N n > n_{ε} . Since δ_{n}^{(k)} > ε_{n}(Ω) , it follows that N(Ω, δ_{n}^{(k)}) ≤ n and then −(d − ε) ln δ_{n}^{(k)} ≤ ln N(Ω, δ_{n}^{(k)}) ≤ ln n . Therefore (δ_{n}^{(k)})^{−(d−ε)} ≤ n and

n^{− \frac{1}{d−ε}} ≤ δ_{n}^{(k)} = (1 + \frac{1}{k}) ε_{n}(Ω) .

By letting k → +∞ we obtain ε_{n}(Ω) ≥ n^{− \frac{1}{d−ε}} ∀n > n_{ε} ■

Corollary 13. If 0 < \overline{dim_{B}}Ω ≤ d , then for all ε > 0 there exists n_{ε} ∈ N so that ε_{n}(Ω) ≤ 2(n + 1)^{− \frac{1}{d+ε}} for n > n_{ε} . PROOF. - We set δ_{n}^{(k)} = (1 − 1/k)φ_{n}(Ω) ∀n, k ∈ N and note that, by proposition 8,

lim sup_{n} \frac{ln M(Ω, δ_{n}^{(k)})}{− ln δ_{n}^{(k)}} ≤ d ∀k ∈ N .

Now we can proceed as in the proof of Corollary 12 and we obtain the desired result ■

Corollary 14. If dim_{B} Ω = d > 0 , then for all ε ∈]0, d[ there exists n_{ε} ∈ N so that n^{− \frac{1}{d−ε}} ≤ ε_{n}(Ω) ≤ 2(n + 1)^{− \frac{1}{d+ε}} for n > n_{ε} .

Example 2. Let p ∈ N and Ω = \{∑_{k=1}^{+∞} \frac{c_{k}}{p^{k}} | c_{k} ∈ \{0, p − 1\} ∀k ∈ N} be the (generalized) Cantor set in R ; by an argument due to C. Richter (personal communication) we see that

ε_{2^{n}+h}(Ω) = \frac{1}{2p^{n}}

for every n ∈ N , h ∈ Z with 0 ≤ h < 2^{n} .

Indeed for all n ∈ N ,

Ω ⊆ Ω_{n} = ⋃_{i_{1}=1}^{2} ⋃_{i_{2}=1}^{2} ... ⋃_{i_{n}=1}^{2} [∑_{k=1}^{n} \frac{(i_{k} − 1)(p − 1)}{p^{k}}, ∑_{k=1}^{n} \frac{(i_{k} − 1)(p − 1)}{p^{k}} + \frac{1}{p^{n}} ]

and

∑_{k=1}^{n} \frac{(i_{k} − 1)(p − 1)}{p^{k}}, ∑_{k=1}^{n} \frac{(i_{k} − 1)(p − 1)}{p^{k}} + \frac{1}{p^{n}} ∈ Ω ∀i_{1}, i_{2}, ..., i_{n} ∈ \{1, 2\} .

Moreover, if h ∈ Z , 0 ≤ h < 2^{n} , then we still have

ε_{2^{n}+h}(Ω) = \frac{1}{2p^{n}}

because the additional h intervals cannot be equally distributed between the 2^{n} connected components of Ω_{n} .

Let d = log_{p} 2 and let n, h be as before; then

ε_{2^{n}+h}(Ω)(2^{n} + h)^{\frac{1}{d}} < ε_{2^{n}+h}(Ω)2^{\frac{n+1}{d}} = \frac{1}{2p^{n}} p^{n+1} = \frac{p}{2} .

On the other hand

ε_{2^{n}+h}(Ω)(2^{n} + h)^{\frac{1}{d}} ≥ ε_{2^{n}+h}(Ω)2^{\frac{n}{d}} = \frac{1}{2p^{n}} p^{n} = \frac{1}{2} .

By combining the two inequalities we obtain

\frac{1}{2}n^{− \frac{1}{d}} ≤ ε_{n}(Ω) < \frac{p}{2}n^{− \frac{1}{d}} ∀n ∈ N .

By Corollary 11 dim_{B} Ω = d .

The constants p/2 and 1/2 are sharp. Indeed lim_{n} ε_{2^{n+1}−1}(Ω)(2^{n+1}− 1)^{1/d} = p/2 and ε_{2^{n}}(Ω)(2^{n})^{1/d} = 1/2 ∀n ∈ N .

3. Entropy numbers of d-sets and of fractals generated by systems of contractions

3.1. d-sets in R^{N}

Proposition 15. Let d ∈]0, N] and Γ ⊆ R^{N} be a bounded d-set. Then

(\frac{μ(Γ)}{c_{2}})^{\frac{1}{d}} n^{− \frac{1}{d}} ≤ ε_{n}(Γ) ≤ 2 (\frac{μ(Γ)}{c_{1}})^{\frac{1}{d}} n^{− \frac{1}{d}} ∀n ∈ N ,

where μ is a d-measure on Γ and c_{1} = c_{1}(|Γ|) , c_{2} = c_{2}(|Γ|) . PROOF. - Let n ∈ N and r ∈]0, φ_{n}(Γ)[ . Then there exist q ∈ N , q > n and y_{1}, y_{2}, ..., y_{q} ∈ Γ so that ||y_{i} − y_{j}||_{2} > 2r ∀i, j ∈ N i < j ≤ q .

For each i , Γ ⋂ B_{R^{N}} (y_{i}, r) is closed and then μ-measurable; moreover, for different i, j , B_{R^{N}} (y_{i}, r) ⋂ B_{R^{N}} (y_{j}, r) = ∅ because ||y_{i} − y_{j}||_{2} > 2r .

By the definition of d-set we have

nc_{1}r^{d} < qc_{1}r^{d} ≤ ∑_{i=1}^{q} μ(B_{R^{N}} (y_{i}, r)) = μ(⋃_{i=1}^{q} B_{R^{N}} (y_{i}, r)) ≤ μ(Γ)

and r^{d} < \frac{μ(Γ)}{nc_{1}} . Then

r < (\frac{μ(Γ)}{c_{1}})^{\frac{1}{d}} n^{− \frac{1}{d}} ∀r ∈]0, φ_{n}(Γ)[

and the second inequality follows from (1).

Now, let ε ∈]ε_{n}(Γ), |Γ|] . Then there exist p ∈ N , p ≤ n and x_{1}, x_{2}, ..., x_{p} ∈ R^{N} so that

Γ ⊆ ⋃_{k=1}^{p} B_{R^{N}} (x_{k}, ε) .

Then

nc_{2}ε^{d} ≥ pc_{2}ε^{d} ≥ ∑_{k=1}^{p} μ(B_{R^{N}} (x_{k}, ε)) ≥ μ(⋃_{k=1}^{p} B_{R^{N}} (x_{k}, ε)) = μ(Γ)

and

ε^{d} ≥ \frac{μ(Γ)}{nc_{2}} ∀ε ∈]ε_{n}(Γ), |Γ|] .

So the first inequality follows ■

Remark 5. Let d ∈]0, N] and Γ ⊆ R^{N} be a bounded d-set. By Proposition 15 and Corollary 11 it follows that dim_{B} Γ = dim Γ =

d .

This fact is also proven, for instance, in [7], chapter 5, theorem 5.7, without using entropy numbers.

Remark 6. It is well known that for a bounded non-empty set Ω ⊆ R^{N} , dim_{B} Ω and dim Ω are not necessarily equal.

We report here a simple example. Let N = 1 and Ω = [0, 1] ⋂ Q ; we have dim Ω = 0 , but dim_{B} Ω = 1 .

Indeed it is easy to see that the restriction of the Lebesgue measure on R to the set [0, 1] satisfies the condition (5) with c_{1} = 1 , c_{2} = 2 and d = 1 . Then, by proposition 15,

\frac{1}{2}n^{−1} ≤ ε_{n}([0, 1]) ≤ 2n^{−1} ∀n ∈ N .

But ε_{n}(Ω) = ε_{n}(\overline{Ω}) = ε_{n}([0, 1]) and by Corollary 11 it follows dim_{B} Ω = 1.

Remark 7. In the hypothesis of Proposition 15 we have

c_{1}|Γ|^{d} ≤ μ(B_{R^{N}} (x_{0}, |Γ|)) ≤ c_{2}|Γ|^{d} ∀x_{0} ∈ Γ

and since μ(B_{R^{N}} (x_{0}, |Γ|)) = μ(Γ) , it follows

(\frac{c_{1}}{c_{2}} )^{\frac{1}{d}} |Γ|n^{− \frac{1}{d}} ≤ ε_{n}(Γ) ≤ 2 (\frac{c_{2}}{c_{1}} )^{\frac{1}{d}} |Γ|n^{− \frac{1}{d}} ∀n ∈ N .

Remark 8. Let 𝔉 = {f_{1}, f_{2}, ..., f_{M}} be a system of contractive similitudes and suppose that 𝔉 satisfies the open set condition. Let O be a corresponding open set and suppose that O contains a ball of radius a_{1} and is contained in a ball of radius a_{2}. Then by Theorem 5 and Proposition 15, we have

(\frac{σa_{1}}{1 + 2a_{2}} )^{\frac{N}{dρ}} n^{− \frac{1}{dρ}} ≤ ε_{n}(K) ≤ 2\frac{|K|}{σ}n^{− \frac{1}{dρ}} ∀n ∈ N .

3.2. Upper estimates for fractals generated by systems of bi-Lipschitz

contractions Let (X, D) be a complete separable metric space and 𝔉 = {f_{1}, f_{2}, ...,

f_{M}} a system of contraction maps on X . Without further conditions on 𝔉 , Theorem 5 is no longer true. However, the first of the two inequalities in (7) still holds.

We refer to Section 2 for the notation.

Lemma 16. Let |K| > 0 . If μ_{𝔉} is the invariant measure defined in Section 2, then

μ_{𝔉}(B_{X}(x_{0}, r)) ≥ (\frac{σ}{|K|})^{d_{ρ}} r^{d_{ρ}} ∀x_{0} ∈ K ∀r ∈]0, |K|] .

PROOF. - We suppose σ > 0 (otherwise the statement is obvious). By iterating (4) n times, we obtain

μ_{𝔉}(A) = ∑_{j_{1}=1}^{M} ∑_{j_{2}=1}^{M} ... ∑_{j_{n}=1}^{M} (∏_{k=1}^{n} ρ_{j_{k}})^{d_{ρ}} μ_{𝔉}(f_{j_{1}j_{2}...j_{n}}^{−1}(A)) ∀A ⊆ X . (11)

Let n ∈ N and i_{1}, i_{2}, ..., i_{n} ∈ {1, 2, ..., M} . Since f_{i_{1}i_{2}...i_{n}} is injective, it admits a left-hand inverse function f_{i_{1}i_{2}...i_{n}}^{−1} : f_{i_{1}i_{2}...i_{n}}(X) → X . It follows that

μ_{𝔉}(f_{i_{1}i_{2}...i_{n}}(K)) = ∑_{j_{1}=1}^{M} ∑_{j_{2}=1}^{M} ... ∑_{j_{n}=1}^{M} (∏_{k=1}^{n} ρ_{j_{k}})^{d_{ρ}} μ_{𝔉}(f_{j_{1}j_{2}...j_{n}}^{−1}(f_{i_{1}i_{2}...i_{n}}(K))) ≥

(∏_{k=1}^{n} ρ_{i_{k}})^{d_{ρ}} μ_{𝔉}(f_{i_{1}i_{2}...i_{n}}^{−1}(f_{i_{1}i_{2}...i_{n}}(K))) = (∏_{k=1}^{n} ρ_{i_{k}})^{d_{ρ}} .

Moreover, by (2), we have

|K| ∏_{k=1}^{n} σ_{i_{k}} ≤ |f_{i_{1}i_{2}...i_{n}}(K)| ≤ |K| ∏_{k=1}^{n} ρ_{i_{k}} .

Now, let x_{0} ∈ K . There exists a sequence (i_{j}) of elements in {1, 2, ..., M}, such that {x_{0}} = ⋂_{n=1}^{+∞} f_{i_{1}i_{2}...i_{n}}(K) . Then x_{0} ∈ f_{i_{1}i_{2}...i_{n}}(K) ∀n ∈ N .

Let r ∈]0, |K|] and n = min{k ∈ N | f_{i_{1}i_{2}...i_{k}}(K) ⊆ B_{X}(x_{0}, r)} . We claim that |K| ∏_{k=1}^{n} ρ_{i_{k}} > σr . Otherwise, we would have

|f_{i_{1}i_{2}...i_{n−1}}(K)| ≤ \frac{1}{σ_{i_{n}}} |f_{i_{1}i_{2}...i_{n}}(K)| ≤ \frac{|K|}{σ_{i_{n}}} ∏_{k=1}^{n} ρ_{i_{k}} ≤ \frac{σr}{σ_{i_{n}}} ≤ r

and f_{i_{1}i_{2}...i_{n−1}}(K) ⊆ B_{X}(x_{0}, r) , since x_{0} ∈ f_{i_{1}i_{2}...i_{n−1}}(K) .

Finally we have

μ_{𝔉}(B_{X}(x_{0}, r)) ≥ μ_{𝔉}(f_{i_{1}i_{2}...i_{n}}(K)) ≥ (∏_{k=1}^{n} ρ_{i_{k}})^{d_{ρ}} > (\frac{σr}{|K|})^{d_{ρ}} ■

It may occur that K ⊆ R^{N} , dim K = d and K is not a d-set, as shown in the following two examples.

Example 3. Let N = 1 , M = 2 , f_{1}(x) = x/2 , f_{2}(x) = 1 .

Then K = {0} ⋃{2^{−n} | n ∈ N ⋃{0}} , dim K = 0 and ℋ^{0}([−1, 1] ⋂ K) = ℋ^{0}(K) = +∞ .

Remark 9. Let us consider a set R ⊆ {0, 1, ..., p−1}×{0, 1, ..., q − 1} , p ≥ q and the system of contractive affinities 𝔉 = {f_{(i,j)}|(i, j) ∈ R} where f_{(i,j)}(x, y) = ((x + i)/p, (y + j)/q) .

Then the invariant set K is called a general Sierpiński carpet.

By means of the formulas given in [8] for the Hausdorff and the box dimension of K we know that dim K = dim_{B} K if and only if one of the following conditions is satisfied:

• p = q (and so K is self-similar)

• for every j ∈ {0, 1, ..., q−1} the number of i such that (i, j) ∈ R can take only one value other than 0 .

If no one of the above conditions is satisfied, by Remark 5, K cannot be a d-set. Note that in this Remark the contractions of 𝔉 are bi-Lipschitz maps and the open set condition is satisfied.

Proposition 17. If σ > 0 , then

ε_{n}(K) ≤ 2\frac{|K|}{σ}n^{− \frac{1}{dρ}} ∀n ∈ N .

PROOF. - Let n ∈ N and r ∈]0, φ_{n}(K)[ . Then there exist q ∈ N q > n , y_{1}, y_{2}, ..., y_{q} ∈ K so that D(y_{i}, y_{j}) > 2r ∀i, j ∈ N i < j ≤ q . Therefore B_{X}(y_{i}, r) ⋂ B_{X}(y_{j}, r) = ∅ .

Moreover, ∀j ∈ N j ≤ q , B_{X}(y_{j}, r) is closed and hence μ_{𝔉}- measurable.

By Lemma 16 we have

n (\frac{σ}{|K|})^{d_{ρ}} r^{d_{ρ}} ≤ ∑_{i=1}^{q} μ_{𝔉}(B_{X}(y_{i}, r)) = μ_{𝔉}(⋃_{i=1}^{q} B_{X}(y_{i}, r)) ≤ μ_{𝔉}(X) = 1

and we can conclude as in Proposition 15 ■ We recall that (X, D) denotes a complete separable metric space.

Proposition 18. If X is a closed subset of R^{N} , then dim K ≤ \overline{dim_{B}}K ≤ d_{ρ} . PROOF. - See [4], Chapter 9, Proposition 9.6.

Corollary 19. If X is a closed subset of R^{N} and \overline{dim_{B}}K > 0 , then for every ε > 0 there exists n_{ε} ∈ N such that

ε_{n}(K) ≤ 2(n + 1)^{− \frac{1}{dρ+ε}} for n > n_{ε} .

PROOF. - The proof follows from Proposition 18 and Corollary 13.

Lemma 20. Let E ⊆ X and let s > 0 . If f : X → X is a mapping and c > 0 is a constant such that D(f(x), f(y)) ≤ cD(x, y) ∀x, y ∈ E , then ℋ^{s}(f(E)) ≤ c^{s}ℋ^{s}(E) . PROOF. - See [4], Chapter 2, Proposition 2.2 and note that the proof works even on a separable metric space.

Lemma 21. Let σ > 0 , |K| > 0 and let d = dim K be the Hausdorff dimension of K . Then

ℋ^{d}(K ⋂ B_{X}(x_{0}, r)) ≥ \frac{ℋ^{d}(K)σ^{d}}{|K|^{d log_{ρ} σ}} r^{d log_{ρ} σ} ∀x_{0} ∈ K ∀r ∈]0, |K|] .

PROOF. - Let x_{0} ∈ K , r ∈]0, |K|] and let (i_{j}) be a sequence in {1, 2, ..., M} such that

{x_{0}} = ⋂_{n=1}^{+∞} f_{i_{1}i_{2}...i_{n}}(K) . (12)

We set

n = min{k ∈ N | f_{i_{1}i_{2}...i_{k}}(K) ⊆ B_{X}(x_{0}, r)} . (13)

The map f_{i_{1}i_{2}...i_{n}} is injective and

D(f_{i_{1}i_{2}...i_{n}}^{−1}(x), f_{i_{1}i_{2}...i_{n}}^{−1}(y)) ≤ \frac{D(x, y)}{∏_{j=1}^{n} σ_{i_{j}}} ∀x, y ∈ f_{i_{1}i_{2}...i_{n}}(K) .

It follows from Lemma 20 that

ℋ^{d}(K) = ℋ^{d}(f_{i_{1}i_{2}...i_{n}}^{−1}(f_{i_{1}i_{2}...i_{n}}(K)) ≤ \frac{ℋ^{d}(f_{i_{1}i_{2}...i_{n}}(K))}{(∏_{j=1}^{n} σ_{i_{j}})^{d}} ≤ \frac{ℋ^{d}(B_{X}(x_{0}, r) ⋂ K)}{(∏_{j=1}^{n} σ_{i_{j}})^{d}}

and therefore

ℋ^{d}(B_{X}(x_{0}, r) ⋂ K) ≥ (∏_{j=1}^{n} σ_{i_{j}})^{d} ℋ^{d}(K) ≥ σ^{nd}ℋ^{d}(K) .

By (12) and (13) it follows that |f_{i_{1}i_{2}...i_{n−1}}(K)| > r and so

ℋ^{d}(B_{X}(x_{0}, r) ⋂ K) > \frac{σ^{nd}ℋ^{d}(K)}{|f_{i_{1}i_{2}...i_{n−1}}(K)|^{d}} r^{d} ≥

\frac{σ^{nd}ℋ^{d}(K)}{(∏_{j=1}^{n−1} ρ_{i_{j}})^{d} |K|^{d}} r^{d} ≥ (\frac{σ}{ρ} )^{(n−1)d} (\frac{σ}{|K|})^{d} ℋ^{d}(K)r^{d} .

If σ = ρ the result is proved.

If σ < ρ then, from (2), r < |f_{i_{1}i_{2}...i_{n−1}}(K)| ≤ |K|ρ^{n−1} and since ρ < 1 , it follows n − 1 < log_{ρ} \frac{r}{|K|} .

Being σ/ρ < 1 , we deduce

ℋ^{d}(B_{X}(x_{0}, r) ⋂ K) > (\frac{σ}{ρ} )^{d log_{ρ} \frac{r}{|K|}} (\frac{r}{|K|})^{d} ℋ^{d}(K)σ^{d} = \frac{σ^{d}ℋ^{d}(K)}{|K|^{d log_{ρ} σ}}r^{d log_{ρ} σ} ■

Proposition 22. Let σ > 0 and d = dim K . Then

ε_{n}(K) ≤ 2\frac{|K|}{ρ}n^{− \frac{1}{d log_{ρ} σ}} ∀n ∈ N .

PROOF. - See the proof of Proposition 17 and use the d-dimensional Hausdorff measure and Lemma 21 instead of the invariant measure and Lemma 16.

3.3. Estimations proved directly

Here we give an upper and a lower estimate for the entropy numbers of the fractal K without using arguments of measure theory.

Proposition 23. Under the hypotheses of theorem 1, if moreover M > 1, we have

ε_{n}(K) ≤ \frac{|K|}{ρ} n^{−_{\frac{1}{log \frac{1}{ρ} M}}} ∀n ∈ N .

PROOF. - Let n, k ∈ N with n > k and let i_{1}, i_{2}, ..., i_{n} ∈ {1, 2, ..., M} . We have

D(x_{i_{1}i_{2}...i_{k}}, x_{i_{1}i_{2}...i_{n}}) = D(f_{i_{1}i_{2}...i_{k}}(x_{i_{1}i_{2}...i_{k}}), f_{i_{1}i_{2}...i_{n}}(x_{i_{1}i_{2}...i_{n}})) ≤

(∏_{j=1}^{k} ρ_{i_{j}}) D(x_{i_{1}i_{2}...i_{k}}, f_{i_{k+1}i_{k+2}...i_{n}}(x_{i_{1}i_{2}...i_{n}})) ≤ |K| ∏_{j=1}^{k} ρ_{i_{j}} ≤ |K|ρ^{k}

because x_{i_{1}i_{2}...i_{k}}, x_{i_{1}i_{2}...i_{n}} ∈ K and f_{i_{k+1}i_{k+2}...i_{n}}(x_{i_{1}i_{2}...i_{n}}) ∈ f_{i_{k+1}i_{k+2}...i_{n}}(K) ⊆ K .

Given x ∈ K , there exists a sequence (i_{j}) in {1, 2, ..., M} such that lim_{n} x_{i_{1}i_{2}...i_{n}} = x .

Then, for all k ∈ N

D(x_{i_{1}i_{2}...i_{k}}, x) ≤ |K|ρ^{k} .

Since there exist M^{k} different sequences i_{1}i_{2}...i_{k} in {1, 2, ..., M} , there exist at most M^{k} distinct points x_{i_{1}i_{2}...i_{k}} ∈ X and then

K ⊆ ⋃_{i_{1}=1}^{M} ⋃_{i_{2}=1}^{M} ... ⋃_{i_{k}=1}^{M} B_{X}(x_{i_{1}i_{2}...i_{k}}, |K|ρ^{k}) ;

so

ε_{M^{k}}(K) ≤ |K|ρ^{k} .

Now, let n, k ∈ N with M^{k−1} ≤ n < M^{k} : we have

ε_{n}(K) ≤ ε_{M^{k−1}}(K) ≤ |K|ρ^{k−1} = \frac{|K|}{ρ}ρ^{k} ≤ \frac{|K|}{ρ}ρ^{log_{M} n} ,

because ρ < 1 . Finally, an easy calculation shows that ρ^{log_{M} n} = n^{−1/ log_{\frac{1}{ρ}} M} ■

Remark 10. If ρ_{i} ≠ ρ_{j} for some i, j ∈ {1, 2, ..., M} , then d_{ρ} < log_{\frac{1}{ρ}} M . So Proposition 17 usually gives a better result than Proposition 23. In particular, if ρ is near to 1 , Proposition 23 can be not interesting.

Otherwise |K|/ρ < 2|K|/σ and then the estimate given by Proposition 23 is better than the one of Proposition 17.

Proposition 24. Let δ = min{D(f_{i}(K), f_{j}(K)) | i, j ∈ N , i < j ≤ M} . If M > 1 and σ > 0 then

ε_{n}(K) ≥ \frac{1}{2}δn^{−_{\frac{1}{log \frac{1}{σ} M}}} ∀n ∈ N .

PROOF. - We suppose δ > 0 , otherwise the statement is obvious.

Let k ∈ N and i_{1}, i_{2}, ..., i_{k}, j_{1}, j_{2}, ..., j_{k} ∈ {1, 2, ..., M} be such that i_{n} ≠ j_{n} for some n ∈ {1, 2, ..., k} ; we set p = min{n ∈ {1, 2, ..., k} | i_{n} ≠ j_{n}} . We consider x_{i_{1}i_{2}...i_{k}} and x_{j_{1}j_{2}...j_{k}} :

D(x_{i_{1}i_{2}...i_{k}}, x_{j_{1}j_{2}...j_{k}}) = D(f_{i_{1}i_{2}...i_{k}}(x_{i_{1}i_{2}...i_{k}}), f_{j_{1}j_{2}...j_{k}}(x_{j_{1}j_{2}...j_{k}})) ≥

(∏_{h=1}^{p−1} σ_{i_{h}}) D(f_{i_{p}}(f_{i_{p+1}i_{p+2}...i_{k}}(x_{i_{1}i_{2}...i_{k}})), f_{j_{p}}(f_{j_{p+1}j_{p+2}...j_{k}}(x_{j_{1}j_{2}...j_{k}}))) ≥

δσ^{p−1} ≥ δσ^{k−1} ,

because f_{i_{p+1}i_{p+2}...i_{k}}(x_{i_{1}i_{2}...i_{k}}), f_{j_{p+1}j_{p+2}...j_{k}}(x_{j_{1}j_{2}...j_{k}}) ∈ K and σ < 1 .

Since there exist M^{k} elements of the form x_{i_{1}i_{2}...i_{k}} (and we have just proved that all are distinct), we have

φ_{M^{k}−1}(K) ≥ \frac{1}{2}δσ^{k−1}

and hence

ε_{M^{k}−1}(K) ≥ \frac{1}{2}δσ^{k−1} .

Let n, k ∈ N with M^{k−1} ≤ n < M^{k} . Since σ^{log_{M} n} ≤ σ^{k−1} , it follows that

ε_{n}(K) ≥ ε_{M^{k}−1}(K) ≥ \frac{1}{2}δσ^{log_{M} n} = \frac{1}{2}δn^{−_{\frac{1}{log \frac{1}{σ} M}}} ■

Remark 11. If the family 𝔉 satisfies the open set condition for an open set O such that f_{i}(\overline{O}) ⋂ f_{j}(\overline{O}) = ∅ for every distinct f_{i}, f_{j} ∈ 𝔉 , then δ > 0 .

This stronger form of the open set condition is called separated open set condition in [9].

Example 4. Let X = R , D(x, y) = |x − y| , M = 2 , f_{1}(x) = x/p , f_{2}(x) = (x + p − 1)/p with p ∈ N p > 2 .

Then K = \{∑_{i=1}^{+∞} \frac{c_{i}}{p^{i}} | c_{i} ∈ \{0, p − 1\} ∀n ∈ N\} is a (generalized) Cantor set.

Here |K| = 1 , ρ = σ = 1/p , δ = (p − 2)/p and by Propositions 24 and 23 we have

\frac{p − 2}{2p}n^{− \frac{1}{log_{p} 2}} ≤ ε_{n}(K) ≤ pn^{− \frac{1}{log_{p} 2}} ∀n ∈ N .

See [1] pag. 194-196 for a direct proof.

The constants (p − 2)/(2p) and p are not sharp (compare with Example 2).

4. β-spaces

Definition 25. A metric space (X, D) is said to be a β-space if for every β ∈]0, 1[ there exist n_{β} ∈ N and c_{β} > 0 satisfying the following condition: for every x ∈ X and for every r ∈]0, c_{β}[ the closed ball B_{X}(x, r) does not contain n_{β} + 1 points x_{1}, x_{2}, ..., x_{n_{β}+1} with d(x_{i}, x_{j}) > 2βr ( 1 ≤ i < j ≤ n_{β} + 1 ).

Lemma 26. Let X be a β-space and β ∈]0, \frac{1}{2}[ . Let n_{β} and c_{β} be as in the previous definition. Then for every x ∈ X , r ∈]0, c_{β}[ and ρ ∈]0, r[ , the closed ball B_{X}(x, r) does not contain n_{β}^{n} + 1 points x_{1}, x_{2}, ..., x_{n_{β}^{n}+1} with D(x_{i}, x_{j}) > 2ρ ( i < j ≤ n_{β} + 1 ) where n = min{k ∈ N | (2β)^{k}r ≤ 2ρ} . PROOF. - Let A_{1} = {x_{1}, x_{2}, ..., x_{n_{1}}} ⊆ B_{X}(x, r) be a maximal subset such that D(x_{i}, x_{j}) > 2βr ∀i, j ∈ N with 1 ≤ i < j ≤ n_{1} . Obviously n_{1} ≤ n_{β} ; moreover

B_{X}(x, r) ⊆ ⋃_{i_{1}=1}^{n_{1}} B_{X}(x_{i_{1}}, 2βr) ,

otherwise A_{1} would not be a maximal subset of B_{X}(x, R) .

Since 2βr < r < c_{β} , we can apply the same argument to B_{X}(x_{i_{1}}, 2βr) for each 1 ≤ i_{1} ≤ n_{1} and by proceeding recursively we obtain that for every n ∈ N the maximum number of points in B_{X}(x, r) with mutual distance larger than (2β)^{n}r is less than or equal to n_{β}^{n} .

Let n ∈ N be so that (2β)^{n}r ≤ 2ρ . Then there are at most n_{β}^{n} points x_{1}, x_{2}, ..., x_{n_{β}^{n}} ∈ B_{X}(x, r) so that D(x_{i}, x_{j}) > 2ρ ∀i, j ∈ N with i < j ≤ n_{β}^{n} ■

Proposition 27. If X is a β-space, then given β ∈]0, \frac{1}{2}[ , x ∈ X, r ∈]0, c_{β}[, we have

ε_{n}(B_{X}(x, r)) ≤ \frac{r}{β} n^{\frac{1}{log_{2β} n_{β}}} .

PROOF. - Let ρ ∈]0, φ_{n}(B_{X}(x, r))[ and k ∈ N with r(2β)^{k+1} ≤ 2ρ < r(2β)^{k}. Then there exist q ∈ N , q > n and x_{1}, x_{2}, ..., x_{q} ∈ B_{X}(x, r) , with D(x_{i}, x_{j}) > 2ρ ∀i, j ∈ N , 1 ≤ i < j ≤ q . By Lemma 26 q ≤ n_{β}^{k+1} , therefore n ≤ n_{β}^{k+1} and k + 1 ≥ log_{n_{β}} n .

We have

ρ < r(2β)^{k} = \frac{r}{2β} (2β)^{k+1} ≤ \frac{r}{2β} (2β)^{log_{n_{β}} n} = \frac{r}{2β} n^{\frac{1}{log_{2β} n_{β}}} ∀ρ ∈]0, φ_{n}(B_{X}(x, r))[

and so

φ_{n}(B_{X}(x, r)) ≤ \frac{r}{2β} n^{\frac{1}{log_{2β} n_{β}}} .

The final result follows from (1) ■

5. Spaces of homogeneous type

Usually in the definition of space of homogeneous type a quasi metric is considered (see for instance [2] pag. 66,67). However, for the sake of simplicity and in accord with the remaining of this note we will give their definition using a metric and not only a quasi metric.

Definition 28. A metric space (X, D) is called a space of homogeneous type if there exists h ∈ N such that every closed ball in X of radius r contains at most h points with mutual distance greater than r/2 .

The number h is called the constant of the space X of homogeneous type.

Remark 12. In [2] is proved that if X is endowed with a Borel measure μ satisfying the condition: there exists c > 0 such that

0 < μ(B_{X}(x, r)) < cμ(B_{X}(x, \frac{r}{2})) < +∞ ∀x ∈ X ∀r ∈]0, +∞[ , (14)

then X is a space of homogeneous type.

Lemma 29. Let r ∈]0, +∞[ and n ∈ N . Then every closed ball in X of radius r contains at most h^{n} points with mutual distance greater than \frac{r}{2^{n}} . PROOF. - See [2], Chapter 3, Lemma 1.1.

Corollary 30. Let Ω ⊆ X be bounded. Then Ω is totally bounded. PROOF. - We suppose that ♯Ω = +∞ , otherwise the statement is obvious. Let ε > 0 , x_{0} ∈ Ω and r > 0 be so that Ω ⊆ B_{X}(x_{0}, r) . We choose A ⊆ Ω such that A ≠ ∅ and D(x, y) > ε ∀x, y ∈ A . By setting n = min\{k ∈ N | \frac{r}{2^{k}} < ε\} we have ♯A ≤ h^{n} by the previous lemma.

Let M = {x_{1}, x_{2}, ..., x_{p}} be a maximal subset of Ω with the previous property. We have

Ω ⊆ ⋃_{i=1}^{p} B_{X}(x_{i}, ε) .

Otherwise there would exist x ∈ Ω with D(x_{i}, x) > ε ∀i ∈ N i ≤ p and then M would not be maximal. ■

Corollary 31. If Ω ⊆ X is bounded, then lim_{n} ε_{n}(Ω) = 0 .

Corollary 32. Every metric space of homogeneous type is a βspace. PROOF. - Let β ∈]0, 1[ and n ∈ N with 2^{n+1}β ≥ 1 . By Lemma 29 the condition of β-space is satisfied with n_{β} = h^{n} and c_{β} = +∞ ■

The proof of the following Proposition is very similar to the one of Proposition 27, but we use Lemma 29 instead of Lemma 26.

Proposition 33. Let x ∈ X . We have

ε_{n}(B_{X}(x, 1)) ≤ 2n^{−1/ log_{2} h} ∀n ∈ N .

Remark 13. Stronger hypotheses are needed in order to have estimates from below for entropy numbers of balls in a space of homogeneous type. Let us consider, for example, the set X = {a, b} with the discrete metric D ; (X, D) is a space of homogeneous type (and also a β-space) and the ”counting measure” is a Borel measure on X satisfying (14). However ε_{n}(X) = 0 ∀n ∈ N\{1} .

Remark 14. Not every β-space is a space of homogeneous type. In fact, let us consider the Hilbert space ℓ^{2} and let X ⊆ ℓ^{2} be

the usual orthonormal basis. The set X , equipped with the metric induced by the ℓ^{2}-norm, is a β-space, but it is not of homogeneous type. Indeed, for every β ∈]0, 1[ , the numbers n_{β} = 1 and c_{β} = \sqrt{2} satisfy the condition of β-space; but each ball with radius r = 2 contains infinitely many points with mutual distance \sqrt{2} > 1 .

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