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Kakeya-Type Sets Over Cantor Sets of Directions in $$\mathbb {R}^{d+1}$$ R d + 1

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Given a Cantor-type subset $\Omega$ of a smooth curve in $\mathbb R^{d+1}$, we construct examples of sets that contain unit line segments with directions from $\Omega$ and exhibit analytical features similar to those of classical Kakeya sets of arbitrarily small $(d+1)$-dimensional Lebesgue measure. The construction is based on probabilistic methods relying on the tree structure of $\Omega$, and extends to higher dimensions an analogous planar result of Bateman and Katz. In particular, the existence of such sets implies that the directional maximal operator associated with the direction set $\Omega$ is unbounded on $L^p(\mathbb{R}^{d+1})$ for all $1\leq p<\infty$.
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arXiv:1404.6235v1 [math.CA] 24 Apr 2014
Kakeya-type sets over Cantor sets of directions in Rd+1
Edward Kroc and Malabika Pramanik
Abstract
Given a Cantor-type subset Ω of a smooth curve in Rd+1, we construct examples
of sets that contain unit line segments with directions from Ω and exhibit analyt-
ical features similar to those of classical Kakeya sets of arbitrarily small (d+ 1)-
dimensional Lebesgue measure. The construction is based on probabilistic methods
relying on the tree structure of Ω, and extends to higher dimensions an analogous
planar result of Bateman and Katz [4]. In particular, the existence of such sets
implies that the directional maximal operator associated with the direction set Ω is
unbounded on Lp(Rd+1) for all 1 p < .
Contents
1 Introduction 2
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Results...................................... 5
2 Overview of the proof of Theorem 1.2 7
2.1 Steps of the proof and layout . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Construction of a Kakeya-type set . . . . . . . . . . . . . . . . . . . . . . 9
3 Geometric Facts 11
4 Rooted, labelled trees 16
4.1 The terminology of trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2 Encoding bounded subsets of the unit interval by trees . . . . . . . . . . . 18
4.3 Encoding higher dimensional bounded subsets of Euclidean space by trees 20
5 Electrical circuits and percolation on trees 22
5.1 The percolation process associated to a tree . . . . . . . . . . . . . . . . . 22
5.2 Trees as electrical networks . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.3 Estimating the survival probability after percolation . . . . . . . . . . . . 25
2010 Mathematics Subject Classification. 28A75, 42B25 (primary), and 60K35 (secondary).
1
6 The random mechanism and the property of stickiness 27
6.1 Slope assignment algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.2 Construction of Kakeya-type sets revisited . . . . . . . . . . . . . . . . . . 30
7 Slope probabilities and root configurations 30
7.1 Four point root configurations . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.2 Three point root configurations . . . . . . . . . . . . . . . . . . . . . . . . 35
8 Proposition 6.4: Proof of the lower bound (2.5) 36
8.1 Proof of Proposition 8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
8.2 Proof of Proposition 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
8.3 Expected intersection counts . . . . . . . . . . . . . . . . . . . . . . . . . 41
9 Proposition 6.4: Proof of the upper bound (2.6) 48
References 51
1 Introduction
1.1 Background
A Kakeya set (also called a Besicovitch set) in Rd+1 is a set that contains a unit line
segment in every direction. The study of such sets spans approximately a hundred
years. The first major analytical result in this area, due to Besicovitch [5], shows that
there exist Kakeya sets with Lebesgue measure zero. Over the past forty-plus years,
dating back at least to the work of Fefferman [10], the study of Kakeya sets has been
a simultaneously fruitful and vexing endeavor. On one hand its applications have been
found in many deep and diverse corners of analysis, PDEs, additive combinatorics and
number theory. On the other hand, certain fundamental questions concerning the size
and dimensionality of such sets have eluded complete resolution.
In order to obtain quantitative estimates for analytical purposes, it is often convenient
to work with the δ-neighborhood of a Kakeya set, rather than the set itself. Here δis
an arbitrarily small positive constant. The δ-neighborhood of a Kakeya set is therefore
an object that consists of many thin δ-tubes. A δ-tube is by definition a cylinder of
unit axial length and spherical cross-section of radius δ. The defining property of a zero
measure Kakeya set dictates that the volume of its δ-neighborhood goes to zero as δ0,
while the sum total of the sizes of these tubes is roughly a positive absolute constant.
Indeed, a common construction of thin Kakeya sets in the plane (see for example [23,
Chapter 10]) relies on the following fact: given any ǫ > 0, there exists an integer N1
and a collection of 2N-tubes, i.e., a family of 1 ×2Nrectangles, {Pt: 1 t2N}in
R2such that [
t
Pt< ǫ, and X
t|e
Pt|= 1.(1.1)
2
Here | · | denotes Lebesgue measure (in this case two-dimensional), and e
Ptdenotes the
“reach” of the tube Pt, namely the tube obtained by translating Ptby two units in
the positive direction along its axis. While it is not known that every Kakeya set in
two or higher dimensions shares a similar feature, the ones that do have found repeated
applications in analysis. Fundamental results have relied on the existence of such sets, for
example the lack of differentiation for integral averages over parallelepipeds of arbitrary
orientation, and the counterexample of the ball multiplier [23, Chapter 10]. The property
described above continues to be the motivation for the Kakeya-type sets that we will
study in the present paper.
Definition 1.1. For d1, we define a set of directions to be a compact subset of
Rd+1. We say that a tube in Rd+1 has orientation ωor a tube is oriented in direction
ωif its axis is parallel to ω. We say that admits Kakeya-type sets if one can find a
constant C01such that for any N1, there exists δN>0,δN0as N→ ∞ and a
collection of δN-tubes {P(N)
t} ⊆ Rd+1 with orientations in with the following property:
if EN:= [
t
P(N)
t, E
N(C0) := [
t
C0P(N)
t,then lim
N→∞ |E
N(C0)|
|EN|=.(1.2)
Here |·|denotes (d+ 1)-dimensional Lebesgue measure, and C0P(N)
tdenotes the tube
with the same centre, orientation and cross-sectional radius as P(N)
t, but C0times its
length. We will refer to {EN:N1}as sets of Kakeya-type.
Specifically in this paper, we will be concerned with certain subsets of a curve, either
on the sphere Sd, or equivalently on a hyperplane at unit distance from the origin, that
admit Kakeya-type sets.
Kakeya and Kakeya-type sets of zero measure have intrinsic structural properties that
continually prove useful in an analytical setting. The most important of these properties
is arguably the so-called stickiness property, originally observed by Wolff [25]. Roughly
speaking, if a Kakeya-type set is a collection of many overlapping line segments, then
stickiness dictates that the map which sends a direction to the line segment in the set
with that direction is almost Lipschitz, with respect to suitably defined metrics. Another
way of expressing this is that if the origins of two overlapping δ-tubes are positioned close
together, then the angle between these thickened line segments must be small, resulting
in the intersection taking place far away from the respective bases. This idea, which has
been formalized in several different ways in the literature [25], [14], [15], [13], will play a
central role in our results, as we will discuss in Section 6.
Geometric and analytic properties of Kakeya and Kakeya-type sets are often studied
using a suitably chosen maximal operator. Conversely, certain blow-up behavior for such
operators typically follow from the existence of such sets. We introduce two such well-
studied operators for which the existence of Kakeya-type sets implies unboundedness.
Given a set of directions Ω, consider the directional maximal operator Ddefined
by
Df(x) := sup
ω
sup
h>0
1
2hZh
h|f(x+ωt)|dt, (1.3)
3
where f:Rd+1 Cis a function that is locally integrable along lines. Also, for any
locally integrable function fon Rd+1, consider the Kakeya-Nikodym maximal operator
Mdefined by
Mf(x) := sup
ω
sup
Px
Pkω
1
|P|ZP|f(y)|dy, (1.4)
where the inner supremum is taken over all cylindrical tubes Pcontaining the point x,
oriented in the direction ω. The tubes are taken to be of arbitrary length land have
circular cross-section of arbitrary radius r, with rl. If Ω is a set with nonempty
interior, then due to the existence of Kakeya sets with (d+ 1)-dimensional Lebesgue
measure zero [5], Dand Mare both unbounded as operators on Lp(Rd+1) for all
1p < . More generally, if Ω admits Kakeya-type sets, then these operators are
unbounded on Lp(Rd+1) for all 1 p < (see Section 1.2 below).
The complementary case when Ω has empty interior has been studied extensively
in the literature. It is easy to see that the operators in (1.3) and (1.4) exhibit a kind
of monotonicity: if Ω , then Df(x)Df(x) and Mf(x)Mf(x), for any
suitable function f. Since these operators are unbounded when Ω= the unit sphere Sd,
treatment of the positive direction – identifying “small” sets of directions Ω for which
these operators are bounded on some Lp– has garnered much attention [20, 6, 22, 2,
1, 21]. These types of results rely on classical techniques in Lp-theory, such as square
function estimates, Littlewood-Paley theory and almost-orthogonality principles.
For a general dimension d1, Nagel, Stein and Wainger [20] showed that Dis
bounded on all Lp(Rd+1 ), 1 < p ≤ ∞, when Ω = {(va1
i,...,vad+1
i) : i1}. Here
0< a1<··· < ad+1 are fixed constants, and {vi:i1}is a sequence obeying
0< vi+1 λvifor some lacunary constant 0 < λ < 1. Carbery [6] showed that Dis
bounded on all Lp(Rd+1), 1 < p ≤ ∞, in the special case when Ω is the set given by the
(d+ 1)-fold Cartesian product of a geometric sequence, namely = {(rk1,...,rkd+1 ) :
k1,...,kd+1 Z+}for some 0 < r < 1. Very recently, Parcet and Rogers [21] generalized
an almost-orthogonality result of Alfonseca [1] to extend the boundedness of Don all
Lp(Rd+1), 1 < p ≤ ∞, for sets Ω that are lacunary of finite order, defined in a suitable
sense. Building upon previous work of Alfonseca, Soria, and Vargas [2], Sj¨ogren and
Sj¨olin [22] and Nagel, Stein and Wainger [20], the recent result of Parcet and Rogers [21]
recovers those of its predecessors.
Aside from this set of positive results with increasingly weak hypotheses, there has
also been much development in the negative direction, pioneered by Bateman, Katz
and Vargas [24, 12, 4, 3]. Of special significance to this article is the work of Bateman
and Katz [4], where the authors establish that Dis unbounded in Lp(R2) for all 1
p < if Ω = {(cos θ, sin θ) : θ∈ C1/3}, where C1/3is the Cantor middle-third set. A
crowning success of the methodology of [4] combined with the aforementioned work in
the positive direction (in particular [1]) is a result by Bateman [3] that gives a complete
characterization of the Lp-boundedness of Dand Min the plane, while also describing
all direction sets Ω that admit planar sets of Kakeya-type. The distinctive feature of
this latter body of work [4, 3] dealing with the negative point of view is the construction
4
of counterexamples using a random mechanism that exploits the property of stickiness.
We too adopt this approach to construct Kakeya-type sets in Rd+1,d2 consisting of
tubes whose orientations lie along certain subsets of a curve on the hyperplane {1}×Rd.
1.2 Results
As mentioned above, Bateman and Katz [4] establish the unboundedness of Dand
Mon Lp(R2), for all p[1,), when Ω = {(cos θ, sin θ) : θ∈ C1/3}by constructing
suitable Kakeya-type sets in the plane. In this paper, we extend their result to the
general (d+ 1)-dimensional setting. To this end, we first describe what we mean by a
Cantor set of directions in (d+ 1) dimensions.
Fix some integer M3. Construct an arbitrary Cantor-type subset of [0,1) as
follows.
Partition [0,1] into Msubintervals of the form [a, b], all of equal length M1.
Among these Msubintervals, choose any two that are not adjacent (i.e., do not
share a common endpoint); define C[1]
Mto be the union of these chosen subintervals,
called first stage basic intervals.
Partition each first stage basic interval into Mfurther (second stage) subintervals
of the form [a, b], all of equal length M2. Choose two non-adjacent second stage
subintervals from each first stage basic one, and define C[2]
Mto be the union of the
four chosen second stage (basic) intervals.
Repeat this procedure ad infinitum, obtaining a nested, non-increasing sequence
of sets. Denote the limiting set by CM:
CM=
\
k=1 C[k]
M.
We call CMageneralized Cantor-type set (with base M).
While conventional uniform Cantor sets, such as the Cantor middle-third set, are special
cases of generalized Cantor-type sets, the latter may not in general look like the former.
In particular, sets of the form CMneed not be self-similar, although the actual sequential
selection criterion leading up to their definition will be largely irrelevant for the content of
this article. It is well-known (see [9, Chapter 4]) that such sets have Hausdorff dimension
at most log 2/log M. By choosing Mlarge enough, we can thus construct generalized
Cantor-type sets of arbitrarily small dimension.
In this paper, we prove the following.
Theorem 1.2. Let CM[0,1] be a generalized Cantor-type set described above. Let
γ: [0,1] → {1} × [1,1]dbe an injective map that satisfies a bi-Lipschitz condition
x, y, c|xy| ≤ |γ(x)γ(y)| ≤ C|xy|,(1.5)
for some absolute constants 0< c < 1< C < . Set Ω = {γ(t) : t∈ CM}. Then
5
(a) the set admits Kakeya-type sets,
(b) the operators Dand Mare unbounded on Lp(Rd+1)for all 1p < .
The condition in Theorem 1.2 that γsatisfies a bi-Lipschitz condition can be weak-
ened, but it will help in establishing some relevant geometry. Throughout this exposition,
it is instructive to envision γas a smooth curve on the plane x1= 1, and we recommend
the reader does this to aid in visualization. Our underlying direction set of interest
Ω = γ(CM) is essentially a Cantor-type subset of this curve.
The main focus of this article, for reasons explained below, is on (a), not on (b).
Indeed, the implication (a) =(b) is well-known in the literature; if f=1EN, where
ENis as in (1.2), then there exists a constant c0=c0(d, C0)>0 such that
minDf(x), Mf(x)c0for xE
N(C0).(1.6)
This shows that
min||D||pp,||M||ppc0|E
N(C0)|
|EN|1
p
,which → ∞ if 1 p < .
On the other hand, the condition (a) of Theorem 1.2 is not a priori strictly necessary in
order to establish part (b) of the theorem. Suppose that {GN:N1}and {e
GN:N
1}are two collections of sets with |e
GN|/|GN| → ∞, enjoying the additional property
that for any point xe
GN, there exists a finite line segment originating at xand pointing
in a direction of Ω, which spends at least a fixed positive proportion of its length in GN.
By an easy adaptation of the argument in (1.6), the sequence of test functions fN= 1GN
would then prove the claimed unboundedness of D. Kakeya-type sets, if they exist,
furnish one such family of test functions with GN=ENand e
GN=E
N.
In [21], Parcet and Rogers construct, for certain examples of direction sets, families of
sets GNthat supply a different class of test functions sufficient to prove unboundedness
of the associated directional maximal operators. Similar constructions could in principle
be applied in our situation as well to establish the unboundedness of directional maximal
operators associated with our sets of interest. However, a set as constructed in [21] is
typically a Cartesian product of a planar Kakeya-type set with a cube, and as such not
of Kakeya-type according to Definition 1.1. In particular, it consists of rectangular par-
allelepipeds with possibly different sidelengths, with these sides not necessarily pointing
in a direction from the underlying direction set Ω, although there are line segments with
orientation from Ω contained within them. Further, in contrast with Definition 1.1, e
GN
need not be obtained by translating GNalong its longest side.
The reason for considering Kakeya-type sets in this paper is twofold. First, they ap-
pear as natural generalizations of a classical feature of planar Kakeya set constructions,
as explained in (1.1). Studying higher dimensional extensions of this phenomenon is of
interest in its own right, and this article provides a concrete illustration of a sparse set
of directions that gives rise to a similar phenomenon. Perhaps more importantly, we
use the special direction sets in this paper as a device for introducing certain machinery
6
whose scope reaches beyond these examples. As discussed in Section 1.1, the problem of
determining a characterization of direction sets Ω that give rise to Lp-bounded maximal
operators Dand Mhas garnered much attention. In [21], Parcet and Rogers obtain
a positive result for such operators to be bounded on all Lebesgue spaces in general
dimensions under certain hypotheses involving lacunarity, and conjecture that their con-
dition is essentially sharp. The counterexamples in [21] mentioned above were furnished
as supporting evidence for this claim. We address this conjecture in [16]. The property
of admittance of Kakeya-type sets in the sense of Definition 1.1 turns out to be a critical
feature of this study, and indeed equivalent to the unboundedness of directional maximal
operators. In addition to the framework introduced in [4], the methods developed in the
present article, specifically the investigation of root configurations and slope probabil-
ities in Sections 7 and 8 are central to the analysis in [16]. While the consideration
of general direction sets in [16] necessarily involves substantial technical adjustments,
many of the main ideas of that analysis can be conveyed in the simpler setting of the
Cantor example that we treat here. As such, we recommend that the reader approach
the current paper as a natural first step in the process of understanding properties of
direction sets that give rise to unbounded directional and Kakeya-Nikodym maximal
operators on Lp(Rd+1).
Acknowledgements
The second author would like to thank Gordon Slade of the Department of Mathematics
at the University of British Columbia for a helpful discussion on percolation theory. The
research was partially supported by an NSERC Discovery Grant.
2 Overview of the proof of Theorem 1.2
2.1 Steps of the proof and layout
The basic structure of the proof is modeled on [4], with some important distinctions
that we point out below. Our goal is to construct a family of tubes rooted on the
hyperplane {0}×[0,1)d, the union of which will eventually give rise to the Kakeya-type
set. The slopes of the constituent tubes will be assigned from Ω via a random mechanism
involving stickiness akin to the one developed by Bateman and Katz [4]. The description
of this random mechanism is in Section 6, with the required geometric and probabilistic
background collected en route in Sections 3, 4 and 5. The essential elements of the
construction, barring the details of the slope assignment, have been laid out in Section
2.2 below. The main estimates leading to the proof of Theorem 1.2 are (2.5) and (2.6) in
Proposition 2.1 in this section. Of these the first, a precise version of which is available
in Proposition 6.4, provides a lower bound of aN=log N/N on the size of the part
of the tubes lying near the root hyperplane. The second inequality, also quantified in
Proposition 6.4, yields an upper bound of bN= 1/N for the portion away from it. The
disparity in the relative sizes of these two parts is the desired conclusion of Theorem 1.2
7
The language of trees was a key element in the random construction of [3, 4]. We
continue to adopt this language, introducing the relevant definitions in Section 4 and
providing some detail on the connection between the geometry of Ω and a tree encoding
it. Specifically, the notion of Bernoulli percolation on trees plays an important role in
the proof of (2.6) with bN= 1/N, as it did in the two-dimensional setting. The higher-
dimensional structure of Ω does however result in minor changes to the argument, and
the general percolation-theoretic facts necessary for handling (2.6) have been compiled
in Section 5. Other probabilistic estimates specific to the random mechanism of Section
6 and central to the derivation of (2.5) are separately treated in Section 7. The proof is
completed in Sections 8 and 9.
Of the two estimates (2.5) and (2.6) necessary for the Kakeya-type construction, the
first is the most significant contribution of this paper. A deterministic analogue of (2.5)
was used in [3, 4], where a similar lower bound for the size of the Kakeya-type set was
obtained for every slope assignment σin a certain measure space. The counting argument
that led to this bound fails to produce the necessary estimate in higher dimensions
and is replaced here by a probabilistic statement that suffices for our purposes. More
precisely, the issue is the following. A large lower bound on a union of tubes follows if
they do not have significant pairwise overlap among themselves; i.e. if the total size of
pairwise intersections is small. In dimension two, a good upper bound on the size of
this intersection was available uniformly in every sticky slope assignment. Although the
argument that provided this bound is not transferable to general dimensions, it is still
possible to obtain the desired bound with large probability. A probabilistic statement
similar to but not as strong as (2.5) can be derived relatively easily via an estimate on
the first moment of the total size of random pairwise intersections. Unfortunately, this is
still not sharp enough to yield the disparity in the sizes of the tubes and their translated
counterparts necessary to claim the existence of a Kakeya-type set. To strengthen the
bound, we need a second moment estimate on the pairwise intersections. Both moment
estimates share some common features; for instance,
- Euclidean distance relations between roots and slopes of two intersecting tubes,
- interplay of the above with the relative positions of the roots and slopes within the
respective trees that they live in, which affects the slope assignments.
However, the technicalities are far greater for the second moment compared to the first.
In particular, for the second moment we are naturally led to consider not just pairs,
but triples and quadruples of tubes, and need to evaluate the probability of obtaining
pairwise intersections among these. Not surprisingly, this probability depends on the
structure of the root tuple within its ambient tree. It is the classification of these root
configurations, computation of the relevant probabilities and their subsequent applica-
tion to the estimation of expected intersections that we wish to highlight as the main
contributions of this article.
8
2.2 Construction of a Kakeya-type set
We now choose some integer M3 and a generalized Cantor-type set CM[0,1) as
described in Section 1.2, and fix these items for the remainder of the article. We also fix
an injective map γ: [0,1] → {1}×[1,1]dsatisfying the bi-Lipschitz condition in (1.5).
These objects then define a fixed set of directions Ω = {γ(t) : t∈ CM} ⊆ {1} × [1,1]d.
Next, we define the thin tube-like objects that will comprise our Kakeya-type set. Fix
an arbitrarily large integer N1, typically much bigger than M. Let {Qt:tTN},
parametrized by the index set TN, be the collection of disjoint d-dimensional cubes
of sidelength MNgenerated by the lattice MNZdin the set {0} × [0,1)d. More
specifically, each Qtis of the form
Qt={0} ×
d
Y
l=1 jl
MN,jl+ 1
MN,(2.1)
for some j= (j1,...,jd)∈ {0,1,··· , M N1}d, so that #(TN) = MN d. For technical
reasons, we also define e
Qtto be the κd-dilation of Qtabout its center point, where κdis
a small, positive, dimension-dependent constant. The reason for this technicality, as well
as possible values of κd, will soon emerge in the sequel, but for concreteness choosing
κd=ddwill suffice.
Recall that the Nth iterate C[N]
Mof the Cantor construction is the union of 2Ndisjoint
intervals each of length MN. We choose a representative element of CMfrom each of
these intervals, calling the resulting finite collection D[N]
M. Clearly dist(x, D[N]
M)MN
for every x∈ CM. Set
N:= γ(D[N]
M),(2.2)
so that dist(ω, N)CM Nfor any ωΩ, with Cas in (1.5).
For any tTNand any ωN, we define
Pt,ω := nr+:re
Qt,0s10C0o,(2.3)
where C0is a large constant to be determined shortly (for instance, C0=ddc1will
work, with cas in (1.5)). Thus the set Pt,ω is a cylinder oriented along ω. Its (vertical)
cross-section in the plane x1= 0 is the cube e
Qt. We say that Pt,ω is rooted at Qt.
While Pt,ω is not strictly speaking a tube as defined in the introduction, the distinction
is negligible, since Pt,ω contains and is contained in constant multiples of δ-tubes with
δ=κd·MN. By a slight abuse of terminology but no loss of generality, we will
henceforth refer to Pt,ω as a tube.
If a slope assignment σ:TNNhas been specified, we set Pt,σ := Pt,σ(t). Thus
{Pt,σ :tTN}is a family of tubes rooted at the elements of an MN-fine grid in
{0} × [0,1)d, with essentially uniform length in tthat is bounded above and below by
fixed absolute constants. Two such tubes are illustrated in Figure 1. For the remainder,
we set
KN(σ) := [
tTN
Pt,σ.(2.4)
9
t1Pt1
Pt2
t2
Figure 1: Two typical tubes Pt1and Pt2rooted respectively at t1and t2
in the {x1= 0}–coordinate plane.
For a certain choice of slope assignment σ, this collection of tubes will be shown
to generate a Kakeya-type set in the sense of Definition 1.1. This particular slope
assignment will not be explicitly described, but rather inferred from the contents of the
following proposition.
Proposition 2.1. For any N1, let ΣNbe a finite collection of slope assignments
from the lattice TNto the direction set N. Every σΣNgenerates a set KN(σ)as
defined in (2.4). Denote the power set of ΣNby PN).
Suppose that N,PN),Pr)is a discrete probability space equipped with the prob-
ability measure Pr, for which the random sets KN(σ)obey the following estimates:
Pr {σ:|KN(σ)[0,1] ×Rd| ≥ aN}3
4,(2.5)
and
Eσ|KN(σ)[C0, C0+ 1] ×Rd| ≤ bN,(2.6)
where C01is a fixed constant, and {aN},{bN}are deterministic sequences satisfying
aN
bN→ ∞,as N→ ∞.
Then admits Kakeya-type sets.
Proof. Fix any integer N1. Applying Markov’s Inequality to (2.6), we see that
Pr {σ:|KN(σ)[C0, C0+ 1] ×Rd| ≥ 4bN}Eσ|KN(σ)[C0, C0+ 1] ×Rd|
4bN1
4,
10
so,
Pr {σ:|KN(σ)[C0, C0+ 1] ×Rd| ≤ 4bN}3
4.(2.7)
Combining this estimate with (2.5), we find that
Pr σ:|KN(σ)[0,1] ×Rd| ≥ aN\σ:|KN(σ)[C0, C0+ 1] ×Rd| ≤ 4bN
Pr |KN(σ)[0,1] ×Rd| ≥ aN+ Pr |KN(σ)[C0, C0+ 1] ×Rd| ≤ 4bN1
3
4+3
41 = 1
2.
We may therefore choose a particular σΣNfor which the size estimates on KN(σ)
given by (2.5) and (2.7) hold simultaneously. Set
EN:= KN(σ)[C0, C0+ 1] ×Rd,so that E
N(2C0+ 1) KN(σ)[0,1] ×Rd.
Then ENis a union of δ-tubes oriented along directions in ΩNΩ for which
|E
N(2C0+ 1)|
|EN|aN
4bN→ ∞,as N→ ∞,
by hypothesis. This shows that Ω admits Kakeya-type sets, per condition (1.2).
Proposition 2.1 proves part (a) of our Theorem 1.2. The implication (a) =(b)
has already been discussed in Section 1.2. The remainder of this paper is devoted to
establishing a proper randomization over slope assignments ΣNthat will then allow us
to verify the hypotheses of Proposition 2.1 for suitable sequences {aN}and {bN}. We
return to a more concrete formulation of the required estimates in Proposition 6.4.
3 Geometric Facts
In this section, we will take the opportunity to establish some geometric facts about
two intersecting tubes in Euclidean space. These facts will be used in several instances
within the proof of Theorem 1.2. Nonetheless they are really general observations that
are not limited to our specific arrangement or description of tubes.
Lemma 3.1. For v1, v2Nand t1, t2TN,t16=t2, let Pt1,v1and Pt2,v2be the tubes
defined as in (2.3). If there exists p= (p1,···, pd+1)∈ Pt1,v1Pt2,v2, then the inequality
cen(Qt2)cen(Qt1) + p1(v2v1)2κddMN,(3.1)
holds, where cen(Q)denotes the centre of the cube Q.
Proof. The proof is described in the diagram below. If p∈ Pt1,v1∩ Pt2,v2, then there
exist x1e
Qt1,x2e
Qt2such that p=x1+p1v1=x2+p1v2, i.e., p1(v2v1) = x1x2.
The inequality (3.1) follows since |xicen(Qti)| ≤ κddMNfor i= 1,2.
11
x1Pt1,v1
Pt2,v2
x2
p
Figure 2: A simple triangle is defined by two rooted tubes, Pt1,v1and Pt2,v2,
and any point pin their intersection.
The inequality in (3.1) provides a valuable tool whenever an intersection takes place.
For the reader who would like to look ahead, the Lemma 3.1 will be used along with
Corollary 3.2 to establish Lemma 8.5. The following Corollary 3.3 will be needed for the
proofs of Lemmas 8.6 and 8.10.
Corollary 3.2. Under the hypotheses of Lemma 3.1 and for κd>0suitably small,
|p1(v2v1)| ≥ κdMN.(3.2)
Proof. Since t16=t2, we must have |cen(Qt1)cen(Qt2)| ≥ MN. Thus an intersection
is possible only if
p1|v2v1| ≥ |cen(Qt2)cen(Qt1)| − 2κddMN(1 2κdd)MNκdMN,
where the first inequality follows from (3.1) and the last inequality holds for an appro-
priate selection of κd.
Corollary 3.3. If t1TN,v1, v2Nand a cube QRd+1 of sidelength C1MNwith
sides parallel to the coordinate axes are given, then there exists at most C2=C2(C1)
choices of t2TNsuch that Pt1,v1∩ Pt2,v2Q6=.
Proof. As p= (p1,···, pd+1) ranges in Q,p1ranges over an interval Iof length C1MN.
If p∈ Pt1,v1∩Pt2,v2Q, the inequality (3.1) and the fact diam(Ω) diam({1[1,1]d) =
2dimplies
cen(Qt2)cen(Qt1) + cen(I)(v2v1)≤ |(p1cen(I))(v2v1)|+ 2κddMN
12
= 2d(C1+κd)MN,
restricting cen(Qt2) to lie in a cube of sidelength 2d(C1+κd)MNcentred at cen(Qt1)
cen(I)(v2v1). Such a cube contains at most C2sub-cubes of the form (2.1), and the
result follows.
A recurring theme in the proof of Theorem 1.2 is the identification of a criterion that
ensures that a specified point lies in the Kakeya-type set KN(σ) defined in (2.4). With
this in mind, we introduce for any p= (p1, p2,···, pd+1)[0,10C0]×Rda set
Poss(p) := Qt:tTN,there exists vNsuch that p∈ Pt,v.(3.3)
This set captures all the possible MN-cubes of the form (2.1) in {0}×[0,1)dsuch that
a tube rooted at one of these cubes has the potential to contain p, provided it is given
the correct orientation. Note that Poss(p) is independent of any slope assignment σ.
Depending on the location of p, Poss(p) could be empty. This would be the case if plies
outside a large enough compact subset of [0,10C0]×Rd, for example. Even if Poss(p) is
not empty, an arbitrary slope assignment σmay not endow any Qtin Poss(p) with the
correct orientation.
In the next lemma, we list a few easy properties of Poss(p) that will be helpful later,
particularly during the proof of Lemma 9.3. Lemma 3.4 establishes the main intuition
behind the Poss(p) set, as we give a more geometric description of Poss(p) in terms of
an affine copy of the direction set ΩN. This is illustrated in Figure 3 for a particular
choice of directions ΩN.
Lemma 3.4. (a) For any slope assignment σ,
Qt:tTN, p Pt,σPoss(p).
(b) For any p[0,10C0]×Rd,
Poss(p) = Qt:e
Qt(pp1N)6=(3.4)
⊆ {Qt:tTN, Qt(pp1N)6=∅}.(3.5)
Note that the set in (3.4) could be empty, but the one in (3.5) is not.
Proof. If pPt,σ, then p∈ Pt,σ(t)with σ(t) equal to some vΩ. Thus Pt,v contains
pand hence QtPoss(p), proving part (a). For part (b), we observe that p∈ Pt,v for
some vNif and only if pp1ve
Qt, i.e., e
Qt(pp1N)6=. This proves the
relation (3.4). The containment in (3.5) is obvious.
We will also need a bound on the cardinality of Poss(p) within a given cube, and on
the cardinality of possible slopes that give rise to indistinguishable tubes passing through
a given point. We now prescribe these. Lemmas 3.5 and 3.6 are not technically needed
for the remainder, but can be viewed as steps toward establishing Lemma 3.7 which will
prove critical throughout Section 9. Not surprisingly, the Cantor-like construction of Ω
plays a role in all these estimates.
13
(a)
p
(b)
Figure 3: Figure (a) depicts the cone generated by a second stage Cantor
construction, 2, on the set of directions given by the curve {(1, t, t2) : 0
tC}in the {1} × R2plane. In Figure (b), a point p= (p1, p2, p3)
has been fixed and the cone of directions has been projected backward from
ponto the coordinate plane, pp12. The resulting Poss(p)set is thus
given by all cubes Qt,tTNsuch that e
Qtintersects a subset of the curve
{(0, p2p1t, p3p1t2) : 0 tC}.
Lemma 3.5. Given C0, C1>0, there exists C2=C2(C0, C1, M, d)>0with the follow-
ing property. Let p= (p1,···, pd+1)(0,10C0]×Rd, and Qbe any cube in {0}×[0,1)d
with sidelength in [M, M +1)for some N1. Then
#Qt:tTN, QtQ6=,dist(Qt, p p1N)C1MNC22N.(3.6)
Proof. Let jZbe the index such that Mjp1< M j+1. By scaling, the left hand
side of (3.6) is comparable to (i.e., bounded above and below by constant multiples of)
the number of p1
1MN-separated points lying in
Q:= xp1
1Q: dist(x, p1
1pN)C1p1
1MN.
But p1
1pN= (1, c)Nis an image of ΩNfollowing an inversion and translation.
This implies that there is a subset Ω
Nof ΩN, depending on pand p1
1Qand with diameter
O(Mj), such that Qis contained in a O(MjN)-neighborhood of
N+ (1, c). The
number of MjN-separated points in Qis comparable to that in Ω
N.
14
Suppose first that j. If C⊆ C[N]
Mis defined by the requirement Ω
N=γ(C),
then (1.5) implies that diam(C) = O(Mj). Thus Cis contained in at most O(1)
intervals of length Mjchosen at step jin the Cantor-type construction. Each
chosen interval at the kth stage gives rise to two chosen subintervals at the next stage,
with their centres being separated by at least Mk1. So the number of MjN-separated
points in C, and hence γ(C) is O(2(Nj)(j)) = O(2N) as claimed. The case j
is even simpler, since the number of MjN-separated points in Cis trivially bounded
by 2Nj2N.
Lemma 3.6. Fix tTNand p= (p1,···, pd+1)[M, M +1]×Rd, for some
0N. Let Qbe a cube centred at pof sidelength C1MN. Then
#vN:Q∩ Pt,v 6=C22.
Proof. If both Pt,v and Pt,vhave nonempty intersection with Q, then there exist q=
(q1,···, qd+1), q= (q
1,···, q
d+1)Qsuch that both qq1vand qq
1vland in e
Qt.
Thus,
p1|vv| ≤ |(qp1v)(qp1v)|+|qq|
≤ |(qq1v)(qq
1v)|+|q1p1||v|+|q
1p1||v|+|qq|
(κdd+ 10C1d)MN.
In other words, |vv| ≤ (10C1+κd)dMN. Recalling that v=γ(α) and v=γ(α)
for some α, α∈ D[N]
M, combining the last inequality with (1.5) implies that |αα| ≤
C2MN. Thus there is a collection of at most O(1) chosen intervals at step Nof
the Cantor-type construction which α(and hence α) can belong to. Since each interval
gives rise to two chosen intervals at the next stage, the number of possible αand hence
vis O(2).
A slight modification of the proof above yields a stronger conclusion, stated below,
when pis far away from the root hyperplane. We will return to this result several times
in the sequel (see for example Lemma 6.3 for a version of it in the language of trees),
and make explicit use of it in Section 9, specifically in the proofs of Lemmas 9.1 and 9.2.
Lemma 3.7. There exists a constant C01with the following properties.
(a) For any p[C0, C0+ 1] ×Rdand tTN, there exists at most one vNsuch that
p∈ Pt,v. In other words, for every Qtin Poss(p), there is exactly one δ-tube rooted
at tthat contains p.
(b) For any pas in (a), and Qt,QtPoss(p), let v=γ(α),v=γ(α)be the two
unique slopes in Nguaranteed by (a) such that p∈ Pt,v ∩ Pt,v. If kis the largest
integer such that Qtand Qtare both contained in the same cube Q⊆ {0} × [0,1)d
of sidelength Mkwhose corners lie in MkZd, then αand αbelong to the same
kth stage basic interval in the Cantor construction.
15
Proof. (a) Suppose v, vNare such that p∈ Pt,v Pt,v. Then pp1vand pp1v
both lie in e
Qt, so that p1|vv| ≤ κddM N. Since p1C0and (1.5) holds, we
find that
|αα| ≤ κdd
cC0
MN< MN,
where the last inequality holds if C0is chosen large enough. Let us recall from the
description of the Cantor-like construction in Section 1.2 that any two basic rth
stage intervals are non-adjacent, and hence any two points in CMlying in distinct
basic rth stage intervals are separated by at least Mr. Therefore the inequality
above implies that both αand αbelong to the same basic Nth stage interval in
C[N]
M. But D[N]
Mcontains exactly one element from each such interval. So α=αand
hence v=v.
(b) If p∈ Pt,v Pt,v, then p1|vv| ≤ diam( e
Qte
Qt)diam(Q) = dMk. Applying
(1.5) again combined with p1C0, we find that |αα| ≤ d
cC0Mk< Mk,for C0
chosen large enough. By the same property of the Cantor construction as used in
(a), we obtain that αand αlie in the same kth stage basic interval in C[k]
M.
4 Rooted, labelled trees
4.1 The terminology of trees
An undirected graph G:= (V,E) is a pair, where Vis a set of vertices and Eis a
symmetric, nonreflexive subset of V × V, called the edge set. By symmetric, here we
mean that the pair (u, v)∈ E is unordered; i.e. the pair (u, v) is identical to the pair
(v, u). By nonreflexive, we mean Edoes not contain the pair (v, v) for any v∈ V.
A path in a graph is a sequence of vertices such that each successive pair of vertices
is a distinct edge in the graph. A finite path (with at least one edge) whose first and last
vertices are the same is called a cycle. A graph is connected if for each pair of vertices
v6=u, there is a path in Gcontaining vand u. We define a tree to be a connected
undirected graph with no cycles.
All our trees will be of a specific structure. A rooted, labelled tree Tis one whose
vertex set is a nonempty collection of finite sequences of nonnegative integers such that
if hi1,...,ini ∈ T , then
(i.) for any k, 0 kn,hi1,...,iki T , where k= 0 corresponds to the empty
sequence, and
(ii.) for every j∈ {0,1,...,in}, we have hi1,...,in1, ji ∈ T .
We say that hi1,...,in1iis the parent of hi1,...,in1, j iand that hi1,...,in1, jiis
the (j+ 1)th child of hi1,...,in1i. If uand vare two sequences in Tsuch that uis a
child of v, or a child’s child of v, or a child’s child’s child of v, etc., then we say that u
16
is a descendant of v(or that vis an ancestor of u), and we write uv(see the remark
below). If u=hi1,...,imi ∈ T ,v=hj1,...,jni T ,mn, and neither unor vis a
descendant of the other, then the youngest common ancestor of uand vis the vertex in
Tdefined by
D(u, v) = D(v, u) := (,if i16=j1
hi1,...,ikiif k= max{l:il=jl}.(4.1)
One can similarly define the youngest common ancestor for any finite collection of ver-
tices.
Remark: At first glance, using the notation uvto denote when uis a descendant of v
may seem counterintuitive, since uis a descendant of vprecisely when vis a subsequence
of u. However, we will soon be identifying vertices of rooted labelled trees with certain
nested families of cubes in Rd. Consequently, as will become apparent in the next two
subsections, uwill be a descendant of vprecisely when the cube associated with uis
contained within the cube associated with v.
We designate the empty sequence as the root of the tree T. The sequence hi1,...,ini
should be thought of as the vertex in Tthat is the (in+ 1)th child of the (in1+
1)th child,..., of the (i1+ 1)th child of the root. All unordered pairs of the form
(hi1,...,in1i,hi1,...,in1, ini) describe the edges of the tree T. We say that the edge
originates at the vertex hi1,...,in1iand that it terminates at the vertex hi1,...,in1, ini.
Note that every vertex in the tree that is not the root is uniquely identified by the edge
terminating at that vertex. Consequently, given an edge e∈ E, we define v(e) to be the
vertex in Vat which eterminates. The vertex hi1,...,ini ∈ T also prescribes a unique
path, or ray, from the root to this vertex:
∅ → hi1i → hi1, i2i → ··· → hi1, i2,...,ini.
We let Tdenote the collection of all rays in Tof maximal (possibly infinite) length.
For a fixed vertex v=hi1,...,imi T , we also define the subtree (of T) generated by
the vertex vto be the maximal subtree of Twith vas the root; i.e. it is the subtree
{hi1,...,im, j1,...,jki ∈ T :k0}.
The height of the tree is taken to be the supremum of the lengths of all the sequences
in the tree. Further, we define the height h(·), or level, of a vertex hi1,...,iniin the
tree to be n, the length of its identifying sequence. All vertices of height nare said to
be members of the nth generation of the root, or interchangeably, of the tree. More
explicitly, a member vertex of the nth generation has exactly nedges joining it to the
root. The height of the root is always taken to be zero.
If Tis a tree and nZ+, we write the truncation of Tto its first nlevels as
Tn={hi1,...,iki ∈ T : 0 kn}.This subtree is a tree of height at most n. A
tree is called locally finite if its truncation to every level is finite; i.e. consists of finitely
17
many vertices. All of our trees will have this property. In the remainder of this article,
when we speak of a tree we will always mean a locally finite, rooted labelled tree, unless
otherwise specified.
Roughly speaking, two trees are isomorphic if they have the same collection of rays.
To make this precise we define a special kind of map between trees that will turn out to
be very important for us later.
Definition 4.1. Let T1and T2be two trees with equal (possibly infinite) heights. Let
σ:T1→ T2; we call σsticky if
for all v∈ T1,h(v) = h(σ(v)), and
uvimplies σ(u)σ(v)for all u, v ∈ T1.
We often say that σis sticky if it preserves heights and lineages.
A one-to-one and onto sticky map between two trees, whose inverse is then auto-
matically sticky, is an isomorphism and the two trees are said to be isomorphic; we will
write T1
=T2. Two isomorphic trees can be treated as essentially identical objects.
4.2 Encoding bounded subsets of the unit interval by trees
The language of rooted labelled trees is especially convenient for representing bounded
sets in Euclidean spaces. This connection is well-studied in the literature. We refer the
interested reader to [19] for more information.
We start with [0,1) R. Fix any positive integer M2. We define an M-adic
rational as a number of the form i/Mkfor some iZ,kZ+, and an M-adic interval
as [i·Mk,(i+ 1) ·Mk). For any nonnegative integer iand positive integer ksuch that
i < Mk, there exists a unique representation
i=i1Mk1+i2Mk2+···+ik1M+ik,(4.2)
where the integers i1,...,iktake values in ZM:= {0,1,...,M 1}. These integers
should be thought of as the “digits” of iwith respect to its base Mexpansion. An
easy consequence of (4.2) is that there is a one-to-one and onto correspondence between
M-adic rationals in [0,1) of the form i/Mkand finite integer sequences hi1,...,ikiof
length kwith ijZMfor each j. Naturally then, we define the tree of infinite height
T([0,1); M) = {hi1,...,iki:k0, ijZM}.(4.3)
The tree thus defined depends of course on the base M; however, if Mis fixed, as it will
be once we fix the direction set Ω = γ(CM) (see Section 1.2), we will omit its usage in
our notation, denoting the tree T([0,1); M) by T([0,1)) instead.
Identifying the root of the tree defined in (4.3) with the interval [0,1) and the vertex
hi1,...,ikiwith the interval [i·Mk,(i+ 1) ·Mk), where iand hi1,...,ikiare related
by (4.2), we observe that the vertices of T([0,1); M) at height kyield a partition of [0,1)
into M-adic subintervals of length Mk. This tree has a self-similar structure: every
18
vertex of T([0,1); M) has Mchildren and the subtree generated by any vertex as the
root is isomorphic to T([0,1); M). In the sequel, we will refer to such a tree as a full
M-adic tree.
Any x[0,1) can be realized as the intersection of a nested sequence of M-adic
intervals, namely
{x}=
\
k=0
Ik(x),
where Ik(x) = [ik(x)·Mk,(ik(x) + 1) ·Mk). The point xshould be visualized as the
destination of the infinite ray
∅ → hi1(x)i → hi1(x), i2(x)i → ··· → hi1(x), i2(x),...,ik(x)i → ·· ·
in T([0,1); M). Conversely, every infinite ray
∅ → hi1i → hi1, i2i → hi1, i2, i3i·· ·
identifies a unique x[0,1) given by the convergent sum
x=
X
j=1
ij
Mj.
Thus the tree T([0,1); M) can be identified with the interval [0,1) exactly. Any subset
E[0,1) is then given by a subtree T(E;M) of T([0,1); M) consisting of all infinite
rays that identify some xE. As before, we will drop the notation for the base Min
T(E;M) once this base has been fixed.
Any truncation of T(E;M), say up to height k, will be denoted by Tk(E;M) and
should be visualized as a covering of Eby M-adic intervals of length Mk. More
precisely, hi1,...,iki ∈ Tk(E;M) if and only if E[i·Mk,(i+ 1) ·Mk)6=, where i
and hi1,...,ikiare related by (4.2).
We now state and prove a key structural result about our sets of interest, the gener-
alized Cantor sets CM.
Proposition 4.2. Fix any integer M3. Define CMas in Section 1.2. Then
T(CM;M)
=T([0,1); 2).
That is, the M-adic tree representation of CMis isomorphic to the full binary tree,
illustrated in Figure 4.
Proof. Denote T=T(CM;M) and T=T([0,1); 2). We must construct a bijective
sticky map ψ:T T . First, define ψ(v0) = v
0, where v0is the root of Tand v
0is the
root of T.
Now, for any k1, consider the vertex hi1, i2,...,iki ∈ T . We know that ijZM
for all j. Furthermore, for any fixed j, this vertex corresponds to a kth level subinterval
19
Figure 4: A pictorial depiction of the isomorphism between a standard
middle-thirds Cantor set and its representation as a full binary subtree of
the full base M= 3 tree.
of C[k]
M. Every such k-th level interval is replaced by exactly two arbitrary (k+1)-th level
subintervals in the construction of C[k+1]
M. Therefore, there exists N1:= N1(hi1,...,iki),
N2:= N2(hi1,...,iki)ZM, with N1< N2, such that hi1,...,ik, ik+1i ∈ T if and only
if ik+1 =N1or N2. Consequently, we define
ψ(hi1, i2,...,iki) = hl1, l2,...,lki ∈ T ,(4.4)
where
lj+1 =(0 if ij+1 =N1(hi1,...,iji),
1 if ij+1 =N2(hi1,...,iji).
The mapping ψis injective by construction and surjectivity follows from the binary
selection of subintervals at each stage in the construction of CM. Moreover, ψis sticky
by (4.4).
The following corollary is an easy consequence of the above and left to the reader.
Corollary 4.3. Recall the definition of D[N]
Mfrom Section 2.2. Then
TN(D[N]
M;M)
=TN([0,1); 2).
Proposition 4.2 and Corollary 4.3 guarantee that the tree encoding our set of direc-
tions will retain a certain binary structure. This fact will prove vital to establishing
Theorem 1.2.
4.3 Encoding higher dimensional bounded subsets of Euclidean space by trees
The approach to encoding a bounded subset of Euclidean space by a tree extends readily
to higher dimensions. For any i=hj1,...,jdi ∈ Zdsuch that i·Mk[0,1)d, we can
20
apply (4.2) to each component of ito obtain
i
Mk=i1
M+i2
M2+···+ik
Mk,
with ijZd
Mfor all j. As before, we identify iwith hi1,...,iki.
Let φ:Zd
M→ {0,1,...,Md1}be an enumeration of Zd
M. Define the full Md-adic
tree
T([0,1)d;M, φ) = nhφ(i1),...,φ(ik)i:k0,ijZd
Mo.(4.5)
The collection of kth generation vertices of this tree may be thought of as the d-fold
Cartesian product of the kth generation vertices of T([0,1); M). For our purposes, it
will suffice to fix φto be the lexicographic ordering, and so we will omit the notation for
φin (4.5), writing simply, and with a slight abuse of notation,
T([0,1)d;M) = nhi1,...,iki:k0,ijZd
Mo.(4.6)
As before, we will refer to the tree in (4.6) by the notation T([0,1)d) once the base M
has been fixed.
By a direct generalization of our one-dimensional results, each vertex hi1,...,ikiof
T([0,1)d;M) at height krepresents the unique M-adic cube in [0,1)dof sidelength Mk,
containing i·Mk, of the form
j1
Mk,j1+ 1
Mk× · ·· × jd
Mk,jd+ 1
Mk.
As in the one-dimensional setting, any x[0,1)dcan be realized as the intersection of a
nested sequence of M-adic cubes. Thus, we view the tree in (4.6) as an encoding of the
set [0,1)dwith respect to base M. As before, any subset E[0,1)dthen corresponds
to a subtree of T([0,1)d;M).
The connection between sets and trees encoding them leads to the following easy
observations that we record for future use in Lemma 9.3.
Lemma 4.4. Let Nbe the set defined in (2.2).
(a) Given N, there is a constant C1>0(depending only on dand C, c from (1.5))
such that for any 1kN, the number of kth generation vertices in TN(ΩN;M)
is C12k.
(b) For any compact set KRd+1, there exists a constant C(K)>0with the following
property. For any x= (x1,···, xd+1)K, and 1kN, the number of kth
generation vertices in TN(E(x); M)is C(K)2k, where E(x) := (xx1N){0}×
[0,1)d.
Proof. There are exactly 2kbasic intervals of level kthat comprise C[k]
M. Under γ, each
such basic interval maps into a set of diameter at most CMk. Since ΩN=γ(D[N]
M)
21
γ(C[k]
M), the number of kth generation vertices in TN(ΩN;M), which is also the number
of kth level M-adic cubes needed to cover ΩN, is at most C12k. This proves (a).
Let Qbe any kth generation M-adic cube such that QN6=. Then on one hand,
(xx1Q)(xx1N)6=; on the other hand, the number of kth level M-adic cubes
covering (xx1Q) is C(K), and part (b) follows.
Notation: We end this section with a notational update. In light of the discussion
above and for simplicity, we will henceforth identify a vertex u=hi1, i2,···, iki ∈
T([0,1)d) with the corresponding cube {0}×ulying on the root hyperplane {0}×[0,1)d.
In this parlance, a vertex t∈ TN([0,1)d) of height Nis the same as a root cube Qt(or
e
Qt) defined in (2.1), and the notation tustands both for set containment as well as
tree ancestry.
5 Electrical circuits and percolation on trees
5.1 The percolation process associated to a tree
The proof of Theorem 1.2 will require consideration of a special probabilistic process
on certain trees called a (bond) percolation. Imagine a liquid that is poured on top of
some porous material. How will the liquid flow - or percolate - through the holes of
the material? How likely is it that the liquid will flow from hole to hole in at least one
uninterrupted path all the way to the bottom? The first question forms the intuition
behind a formal percolation process, whereas the second question turns out to be of
critical importance to the proof of Theorem 1.2; this idea plays a key role in establishing
the planar analogue of that theorem in Bateman and Katz [4], and again in the more
general framework of [3].
Although it is possible to speak of percolation processes in far more general terms
(see [11]), we will only be concerned with a percolation process on a tree. Accordingly,
given some tree Twith vertex set Vand edge set E, we define an edge-dependent Bernoulli
(bond) percolation process to be any collection of random variables {Xe:e∈ E}, where
Xeis Bernoulli(pe) with pe<1. The parameter peis called the survival probability
of the edge e. We will always be concerned with a particular type of percolation on
our trees: we define a standard Bernoulli(p)percolation to be one where the random
variables {Xe:e∈ E} are mutually independent and identically distributed Bernoulli(p)
random variables, for some p < 1. In fact, for our purposes, it will suffice to consider
only standard Bernoulli( 1
2) percolations.
Rather than imagining a tree with a percolation process as the behaviour of a liquid
acted upon by gravity in a porous material, it will be useful to think of the percolation
process as acting more directly on the mathematical object of the tree itself. Given some
percolation process on a tree T, we will think of the event {Xe= 0}as the event that
we remove the edge efrom the edge set E, and the event {Xe= 1}as the event that we
retain this edge; denote the random set of retained edges by E. Notice that with this
interpretation, after percolation there is no guarantee that E, the subset of edges that
remain after percolation, defines a subtree of T. In fact, it can be quite likely that the
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subgraph that remains after percolation is a union of many disconnected subgraphs of
T.
For a given edge e∈ E, we think of p= Pr(Xe= 1) as the probability that we retain
this edge after percolation. The probability that at least one uninterrupted path remains
from the root of the tree to its bottommost level is given by the survival probability of
the corresponding percolation process. More explicitly, given a percolation on a tree T,
the survival probability after percolation is the probability that the random variables
associated to all edges of at least one ray in Ttake the value 1; i.e.
Pr (survival after percolation on T) := Pr [
RT\
e∈E∩R{Xe= 1}!.(5.1)
Estimation of this probability will prove to be a valuable tool in the proof of Theorem 1.2.
This estimation will require reimagining a tree as an electrical network.
5.2 Trees as electrical networks
Formally, an electrical network is a particular kind of weighted graph. The weights
of the edges are called conductances and their reciprocals are called resistances. In his
seminal works on the subject, Lyons visualizes percolation on a tree as a certain electrical
network. In [17], he lays the groundwork for this correspondence. While his results hold
in great generality, we describe his results in the context of standard Bernoulli percolation
on a locally finite, rooted labelled tree only. We briefly review the concepts relevant to
our application here.
A percolation process on the truncation of any given tree Tis naturally associated
to a particular electrical network. To see this, we truncate the tree Tat height Nand
place the positive node of a battery at the root of TN. Then, for every ray in TN, there
is a unique terminating vertex; we connect each of these vertices to the negative node of
the battery. A resistor is placed on every edge eof TNwith resistance Redefined by
1
Re
=1
1peY
∅⊂v(e)v(e)
pe.(5.2)
Notice that the resistance for the edge eis essentially the reciprocal of the probability
that a path remains from the root of the tree to the vertex v(e) after percolation. For
standard Bernoulli(1
2) percolation, we have
Re= 2h(v(e))1.(5.3)
One fact that will prove useful for us later is that connecting any two vertices at
a given height by an ideal conductor (i.e. one with zero resistance) only decreases the
overall resistance of the circuit. This will allow us to more easily estimate the total
resistance of a generic tree.
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Proposition 5.1. Let TNbe a truncated tree of height Nwith corresponding electrical
network generated by a standard Bernoulli(1
2)percolation process. Suppose at height
k < N we connect two vertices by a conductor with zero resistance. Then the resulting
electrical network has a total resistance no greater than that of the original network.
Proof. Let uand vbe the two vertices at height kthat we will connect with an ideal
conductor. Let R1denote the resistance between uand D(u, v), the youngest common
ancestor of uand v; let R2denote the resistance between vand D(u, v). Let R3denote
the total resistance of the subtree of TNgenerated by the root uand let R4denote the
total resistance of the subtree of TNgenerated by the root v. These four connections
define a subnetwork of our tree, depicted in Figure 5(a). The connection of uand v
by an ideal conductor, as pictured in Figure 5(b), can only change the total resistance
of this subnetwork, as that action leaves all other connections unaltered. It therefore
suffices to prove that the total resistance of the subnetwork comprising of the resistors
R1,R2,R3and R4can only decrease if uand vare joined by an ideal conductor.
(a)
D(u, v)
u v
+
R1R2
R3R4
(b)
D(u, v)
uv
+
R1R2
R3R4
Figure 5: (a) The original subnetwork with the resistors R1,R3and R2,R4
in series; (b) the new subnetwork obtained by connecting vertices uand vby
an ideal conductor.
In the original subnetwork, the resistors R1and R3are in series, as are the resistors
R2and R4. These pairs of resistors are also in parallel with each other. Thus, we
calculate the total resistance of this subnetwork, Roriginal:
Roriginal =1
R1+R3
+1
R2+R41
=(R1+R3)(R2+R4)
R1+R2+R3+R4
.(5.4)
After connecting vertices uand vby an ideal conductor, the structure of our subnetwork
is inverted as follows. The resistors R1</