TWO INFINITE VERSIONS OF NONLINEAR DVORETZKY’S
THEOREM
KEI FUNANO
Abstract. We investigate nonlinear Dvoretzky’s theorem for countably infinite metric
spaces and analytic sets whose Hausdorff dimension are infinite.
1. Introduction and the statement of the results
We say that a metric space X is embedded with distortion D ≥ 1 in a metric space Y
if there exist a map f : X → Y and a constant r > 0 such that
r dX (x, y) ≤ dY (f (x), f (y)) ≤ Dr dX (x, y) for all x, y ∈ X.
Such a map f is called a D-embbeding.
Dvoretzky’s theorem states that for every ε > 0, every n-dimensional normed space
contains a k(n, ε)-dimensional subspace that embeds into a Hilbert space with distortion
1 + ε ([4]). This theorem was conjectured by Grothendieck ([5]). See [10, 11] and [12],
[14, 15] for the estimate of k(n, ε) and the further direction of this theorem.
Bourgain, Figiel, and Milman proved the following theorem as a natural nonlinear
variant of Dvoretzky’s theorem.
Theorem 1.1 (cf. [2]). There exists two universal constants c1 , c2 > 0 satisfies the following. For every ε > 0 every finite metric space X contains a subset S which embeds
into a Hilbert space with distortion 1 + ε and
c1 ε
log |X|.
|S| ≥
log(c2 / ε)
See [1], [8], [13] for the further investigation. It is natural to ask the case where |X| = ∞
in the above theorem. In this paper we prove the following.
Theorem 1.2. For every ε > 0, every countable infinite metric space X has an infinite
subset which embeds into an ultrametric space with distortion 1 + ε.
Recall that a metric space (U, ρ) is called an ultrametric space if for every x, y, z ∈ X we
have ρ(x, y) ≤ max{ρ(x, z), ρ(z, y)}. Since every separable ultrametric space isometrically
Date: November 7, 2011.
2000 Mathematics Subject Classification. 53C21, 53C23.
Key words and phrases. Dvoretzky’s theorem, ultrametric space.
This work was partially supported by Grant-in-Aid for Research Activity (startup) No.23840020.
1
2
KEI FUNANO
embeds into a Hilbert space ([17]), we verify that Theorem 1.1 holds in the case where
|X| = ∞.
Recently Mendel and Naor proved an another variant of Dvoretzky’s theorem, answering
a question by T. Tao ([9]). For a metric space X we denote its Hausdorff dimension by
dimH (X). A subset of a complete separable metric space is called an analytic set if it is
an image of a complete separable metric space under a continuous map.
Theorem 1.3 (cf. [9, Theorem 1.7]). There exists a universal constant c ∈ (0, ∞) such
that for every ε ∈ (0, ∞), every analytic set X whose Hausdorff dimension is finite has a
closed subset S ⊆ X that embeds with distortion 2 + ε in an ultrametric space, and
cε
dimH (S) ≥
dimH (X).
log(1/ ε)
In [9] Mendel and Naor stated the above theorem only for compact metric spaces. As
they remarked in [9, Introduction], their theorem is valid for more general metric spaces.
For example the above theorem holds for every analytic set X since the problem reduce
to a compact subset of X with the same Hausdorff dimension (see [3], [6, Corollary 7]).
In the following theorem we consider the case where dimH (X) = ∞.
Theorem 1.4. For every ε ∈ (0, ∞), every analytic set X whose Hausdorff dimension is
infinite has a closed subset S such that S can be embedded into an ultrametric space with
distortion 2 + ε and has infinite Hausdorff dimension.
It turns out that the proof in [9] is valid in the case where X is a compact metric space
and dimH (X) = ∞ (see Section 3). It follows from the proof of Theorem 1.3 in [9] that if
dimH (X) = ∞, then X contains an arbitrary large dimensional closed subset. Combining
Theorem 1.3 with Theorem 1.4 we find that nonlinear Dvoretzky’s theorem holds for all
analytic sets.
The following theorem due to Mendel and Naor asserts that we cannot replace the
distortion strictly less than 2 in Theorems 1.3 and 1.4.
Theorem 1.5 (cf. [9, Theorem 1.8]). For every α > 0 there exists a compact metric
space (X, d ) of Hausdorff dimension α, such that if S ⊆ X embeds into Hilbert space with
distortion strictly smaller than 2 then dimH (S) = 0.
The case where α = ∞ easily follows from Theorem 1.5.
Tsirel’son constructed an infinite dimensional reflexive Banach space which does not
contain any subspaces which are linealy homeomorphic to one of ℓp (1 ≤ p < +∞) or c0
([16]). In particular, an infinite dimensional analogue of Dvoretzky’s theorem is no longer
true in the linear setting. In contrast to his example, Theorem 1.4 asserts that an infinite
dimensional Dvoretzky’s theorem holds in the nonlinear setting.
2. Proof
We use the following somewhat basic lemma.
TWO INFINITE VERSIONS OF NONLINEAR DVORETZKY’S THEOREM
3
Lemma 2.1. Let X be a separable metric space such that dimH (X) = ∞. Then there exists a sequence {Ki }∞
i=1 of mutually disjoint closed subsets of X such that limi→∞ diam Ki =
0 and limi→∞ dimH Ki = ∞.
Proof. For every x ∈ X we take a closed neighborhood Kx of x such that diam Kx ≤ 1.
Since X is separable, applying
the Lindelöf covering
theorem we get a countable subset
∪
∪
F ⊆ X such that X = x∈F Kx . Since dimH ( x∈F Kx ) = supx∈F dimH Kx , there exist
x1 ∈ F such that dimH Kx1 = ∞ or we can take a sequence {yi }∞
i=1 ⊆ F such that
{dimH Kyi }∞
is
strictly
increasing
and
lim
dim
K
=
∞.
i→∞
H
yi
i=1
We first consider the latter case. We put K1 := Ky1 . By the monotonicity of Kyi we have
∪
∪i−1
dimH (Kyi \ i−1
K
)
=
dim
K
for
i
≥
2.
Dividing
K
\
y
H
y
y
j
i
i
j=1
j=1 Kyj into countable closed
∪i−1
subsets of an arbitrary small diameter, we thus find a closed subset Ki ⊆ Kyi \ j=1
Kyj
∞
such that dimH Ki = dimH Kyi and diam Ki ≤ 1/i. This {Ki }i=1 is a desired sequence.
We consider the former case. Dividing Kx1 into countable closed subsets {Ky1 }y∈F1 so
that diam Ky1 ≤ 2−1 diam Kx1 , we find x2 ∈ F1 such that dimH (Kx12 ) = ∞. Inductively we
find a chain Kx12 ⊇ Kx23 ⊇ Kx34 ⊇ · · · of closed subsets
of X such that dimH (Kxi−1
)=∞
i
∪∞
i
−1
i−1
i−1
j−1
and diam Kxi+1 ≤ 2 diam Kxi . Since Kxi \ j=i (Kxj \ Kxj j+1 ) consists of one point,
we get lim supi→∞ dimH (Kxi−1
\ Kxi i+1 ) = ∞. By taking a subsequence we may assume
i
that limi→∞ dimH (Kxi−1
\Kxi i+1 ) = ∞. Taking a closed subset Ki ⊆ Kxi−1
\Kxi i+1 such that
i
i
dimH Ki ≥ 2−1 dimH (Kxi−1
\ Kxi i+1 ) we easily see that this {Ki }∞
i=1 is a desired sequence.
i
This completes the proof.
We first prove Theorem 1.4. It turns out that Theorem 1.2 follows from the proof of
Theorem 1.4.
Proof of Theorem 1.4. We take a sequence {Ki }∞
i=1 of closed subsets of X in Lemma 2.1.
For each i we fix an element xi ∈ Ki . Note that closed subsets of analytic sets are also
analytic sets. According to Theorem 1.3 there exist Ai ⊆ Ki such that limi→∞ dimH Ai =
∞ and Ai embeds into some ultrametric space (Ui , ρi ) with distortion 2 + ε, i.e., there
exist fi : Ai → Ui satisfying
(2.1)
d (x, y) ≤ ρi (fi (x), fi (y)) ≤ (2 + ε) d (x, y) for any x, y ∈ Ai .
We devide the proof into three cases.
Case 1. {xi }∞
i=1 is not bounded.
By taking a subsequence we may assume that limn→∞ d (x1 , xi ) = ∞ and diam Ki ≤
1/(2 + ε). By taking a subsequence we may also assume that
√
√
√
√
{
1+ε−1
1 + ε − 1 + 2−1 ε 1 + 2−1 ε − 1 }
√
√
(2.2) 1 ≤ min √
,
,
d (A1 , A2 )
2
1 + ε 1 + 2−1 ε
1 + 2−1 ε
and
√
√
√
{
1+ε−1
1 + ε − 1 + 2−1 ε }
√
√
(2.3)
,
d (A1 , Ai ).
d (A1 , Ai−1 ) ≤ min √
1 + ε 1 + 2−1 ε
1 + 2−1 ε
4
KEI FUNANO
for any i ≥ 2. Put Ri := d (Ai , A1 ) for i ≥ 2. Note that diam fi (Ai ) ≤ 1 since fi satisfies
(2.1) and diam Ai ≤ diam Ki ≤ 1/(2 + ε).
For each i ≥ 2 we add an additional point ui,0 to fi (Ai ) and put Yi := fi (Ai ) ∪ {ui,0 }.
Define the distance function ρ̃i on Yi as follows: ρ̃i (u, ui,0 ) := Ri for u ∈ fi (Ai ) and
ρ̃i (u, v) := ρi (u, v) for u, v ∈ fi (Ai ). Since diam fi (Ai ) ≤ 1 ≤ Ri , each (Yi , ρ̃i ) is an
ultrametric space. Let us consider the space
U := {(ui ) ∈
∞
∏
Yi | ui ̸= ui,0 only for finitely many i}
i=2
and define the distance function ρ on U by
ρ((ui ), (vi )) := sup ρ̃i (ui , vi ).
i
It is easy to verify that (U, ρ) is an ultrametric space. For each x ∈ Ai we put
f (x) := (u2,0 , u3,0 , · · · , ui−1,0 , fi (x), ui+1,0 , ui+2,0 , · · · ).
∪∞
We shall prove that f is a (2 + ε)-embedding
from
the
closed
subset
i=2 Ai ⊆ X to the
∪∞
ultrametric space (U, ρ). Note that dimH ( i=2 Ai ) = ∞.
We take two arbitrary points x ∈ Ai and y ∈ Aj (i < j) and fix z ∈ X. By (2.2) and
(2.3), we get
√
d (x, z) ≤ Ri + diam A1 + diam Ai ≤ Ri + 2 ≤ 1 + 2−1 εRi .
Combining this inequality with (2.2) and (2.3) also implies
√
1
1
Rj = √
ρ(f (x), f (y)).
d (x, y) ≥ d (y, z) − d (x, z) ≥ Rj − 1 + 2−1 εRi ≥ √
1+ε
1+ε
and
d (x, y) ≤ d (x, z) + d (y, z) ≤
≤
=
√
√
√
1 + 2−1 εRi +
√
1 + 2−1 εRj
1 + εRj
1 + ερ(f (x), f (y)).
Hence f is a (2 + ε)-embedding.
Case 2. {xi }∞
i=1 is bounded but not totally bounded.
By taking a subsequence, we may assume that there exist two constants c1 , c2 > 0 such
that
c1 ≤ d (xi , xj ) ≤ c2 for any distinct i, j.
∪
For any δ > 0 we divide [c1 , c2 ] = m
j=1 Ij so that diam Ij < δ for any j.
Pick j1 ∈ {1, 2, · · · , m} such that d (xi , x1 ) ∈ Ij1 holds for infinitely many i. Put
X1 := {xi | d (xi , x1 ) ∈ Ij1 } = {xk1 (1) , xk1 (2) , · · · }.
TWO INFINITE VERSIONS OF NONLINEAR DVORETZKY’S THEOREM
5
We then choose j2 ∈ {1, 2, · · · , m}) so that d (xk1 (i) , xk1 (1) ) ∈ Ij2 holds for infinitely many
i and put
X2 := {xk1 (i) ∈ X1 | d (xk1 (i) , xk1 (1) ) ∈ Ij2 } = {xk2 (1) , xk2 (2) , · · · }.
Repeatedly we obtain a sequence {ji }∞
i=1 ⊆ {1, 2, · · · , m} and Xi = {xki (1) , xki (2) , · · · }.
∞
By a pigeon hole argument we find a subsequence {jh(i) }∞
i=1 ⊆ {ji }i=1 which is monochromatic, i.e., jh(i) ≡ l for some l ∈ {1, 2, · · · , m}. We then get d (xkh(i) (i) , xkh(j) (j) ) ∈ Il .
Since diam Il < δ and limi→∞ diam Ai = 0, by choosing sufficiently small δ and taking a
subsequence, we thereby get the following: There exists a number α ≥ c1 such that
α ≤ d (u, v) ≤ (1 + ε)α for any u ∈ Ai and v ∈ Aj (i ̸= j)
and diam Ai ≤ (2 + ε)−1 α. As in Case 1 we add an additional point ui,0 to fi (Ai ) and put
Yi := fi (Ai )∪{ui,0 }. We define the distance function ρ̃i on Yi as follows: ρ̃i (u, ui,0 ) := α for
u ∈ fi (Ai ) and ρ̃i (u, v) := ρi (u, v) for u, v ∈ fi (Ai ). Since diam fi (Ai ) ≤ (2 + ε) diam Ai ≤
α, each (Yi , ρ̃i ) is an ultrametric space. Consider the space
U := {(ui ) ∈
∞
∏
Yi | ui ̸= ui,0 only for finitely many i.}
i=1
with the distance function ρ defined by ρ((ui ), (vi )) := supi ρ̃i (ui , vi ). Then (U, ρ) is a
ultrametric space and the map
f (u) := (u1,0 , u2,0 , · · · , ui−1,0 , fi (u), ui+1,0 , ui+2,0 , · · · )
∪
is a (2 + ε)-embedding from the closed subset ∞
i=1 Ai ⊆ X to the space (U, ρ).
Case 3. {xi }∞
i=1 is totally bounded.
From the totally boundedness, by taking a subsequence, we may assume that {xi }∞
i=1 is
a Cauchy sequence. Since limi→∞ diam Ai = 0, the sequence {Ai }∞
Hausdorff
converges
i=1
to a point x∞ . Note that x∞ ̸∈ Ai for any sufficiently large i since Ai are mutually disjoint
closed subsets of X. Hence, by taking a subsequence, we may also assume that the ratio
d (Ai−1 , x∞ )/ d (Ai , x∞ ) is sufficiently large for each i. Dividing Ai into sufficiently small
pieces as in the proof of Lemma 2.1 allows us to assume that the ratio d (Ai , x∞ )/ diam Ai
is sufficiently ∪
large for every i. Therefore, as in the proof of Case 1 we can embed the
closed subset ∞
i=1 Ai ⊆ X into an ultrametric space with distortion 2 + ε.
Proof of Theorem 1.2. Let X := {x1 , x2 , · · · }. Apply the proof of Theorem 1.4 by identifying each xi with Ki . Note that the loss of the distortion in the proof only comes from
(2.1), which we can ignore in the case where Ai = xi . Hence the space X can embed into
an ultrametric space with distortion 1 + ε. This completes the proof.
3. The case of compact metric spaces
In this section we remark that the case where X is a compact metric space and
dimH (X) = ∞ in Theorem 1.4 also follows from the same method in [9]. Although
6
KEI FUNANO
this section goes along the same line in [9] I explain it precisely for the completeness of
this paper.
Mendel and Naor proved the following theorem in [9]: A metric measure space is a triple
(X, d , µ) consisting of a complete separable metric space (X, d ) and a Borel probability
measure µ on X.
Theorem 3.1 (cf. [9, Theorem 1.9]). There exists a universal constant c ∈ (0, ∞) such
that for every ε ∈ (0, 1/2) there exists cε ∈ (0, ε) with the following property. Every
compact metric measure space (X, d , µ) has a closed subset S ⊆ X such that (S, d ) embeds
into an ultrametric space with distortion 2 + ε, and for every {xi }i∈I ⊆ X and {ri }i∈I ⊆
[0, ∞) such that the balls {B(xi , ri )}i∈I cover S, i.e.,
∪
S⊆
B(xi , ri ),
i∈I
we have
∑
(3.1)
cε
µ(B(xi , cε ri )) log(1/ ε) ≥ 1.
i∈I
As explained in [9] Theorem 3.1 implies Theorem 1.3 (see the following discussion).
Theorem 3.1 also implies Theorem 1.4 in the case where X is a compact metric space
and dimH (X) = ∞ as follows: Fix ε ∈ (0, 1). By Frostman’s lemma (cf. [6], [7]) for any
n ∈ N there exist a Borel probability measure µn on X and a constant Kn > 0 such
that µn (B(x, r)) ≤ Kn rn for any x ∈ X and r > 0. By virtue of Theorem 3.1, there
exists a closed subset Sn ⊆ X such that (Sn , d ) embeds into an ultrametric space with
distortion 2 + ε and Sn satisfies the covering condition in Theorem 3.1 with respect to
the measure µn . Since the compactness of X is inherited to the space of closed subsets
of X with the Hausdorff distance function dH (see e.g. [7]), there exists a subsequence
∞
{Si(n) }∞
n=1 ⊆ {Sn }n=1 and a closed subset S ⊆ X such that limn→∞ dH (Si(n) , S) = 0.
Then as in the proof of [9, Lemma 3.1] the closed subset S embed into an ultrametric
space with distortion 2 + ε.
We shall prove that dimH (S) = ∞. Recall
∑that αfor any α ≥ 0 the α-Hausdorff content
α
H∞ (S) of S is defined as the infimum of j∈J rj of all covers {B(xj , rj )}j∈J of S by
α
open balls and dimH (S) = inf{α ≥ 0 | H∞
(S) = 0} ([7]). Hence it suffices to prove that
α
H∞ (S) > 0 for any α ≥ 0. Let {B(xi , ri )}i∈J be a family of open
∑ balls that cover X.
Since S is compact and limn→∞ dH (Si(n) , S) = 0, we have Si(n) ⊆ j∈J B(xj , rj ) for any
sufficiently large n ∈ N. By the covering condition (3.1) we thereby get
∑
∑
cε
cε
i(n) log(1/
ε)
)
(Ki(n) ci(n)
.
1≤
µi(n) (B(xj , cε rj )) log(1/ ε) ≤
ε rj
j∈J
j∈J
Putting δ := c ε / log(1/ ε) we hence get
∑ δi(n)
≥
rj
j∈J
1
δi(n)
δ
Ki(n)
cε
TWO INFINITE VERSIONS OF NONLINEAR DVORETZKY’S THEOREM
7
i.e.,
i(n)δ
H∞
(S) ≥
1
δi(n)
δ
Ki(n)
cε
> 0,
which implies dimH (S) = ∞.
Acknowledgements. The author would like to express his thanks to Mr. Takumi Yokota
for his suggestion regarding the two theorems in this paper and Mr. Ryokichi Tanaka for
discussion. The author also thanks to Professor Manor Mendel and Professor Assaf Naor
for their useful comments.
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KEI FUNANO
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502
JAPAN
E-mail address: [email protected]