Content uploaded by Edward Kroc

Author content

All content in this area was uploaded by Edward Kroc on Feb 18, 2020

Content may be subject to copyright.

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 δ→0,

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 N≥1

and a collection of 2−N-tubes, i.e., a family of 1 ×2−Nrectangles, {Pt: 1 ≤t≤2N}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 d≥1, we deﬁne a set of directions Ωto be a compact subset of

Rd+1. We say that a tube in Rd+1 has orientation ω∈Ωor a tube is oriented in direction

ωif its axis is parallel to ω. We say that Ω admits Kakeya-type sets if one can ﬁnd a

constant C0≥1such that for any N≥1, there exists δN>0,δN→0as N→ ∞ and a

collection of δN-tubes {P(N)

t} ⊆ Rd+1 with orientations in Ωwith 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:N≥1}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 DΩdeﬁned

by

DΩf(x) := sup

ω∈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

MΩdeﬁned by

MΩf(x) := sup

ω∈Ω

sup

P∋x

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 r≤l. If Ω is a set with nonempty

interior, then due to the existence of Kakeya sets with (d+ 1)-dimensional Lebesgue

measure zero [5], DΩand MΩare both unbounded as operators on Lp(Rd+1) for all

1≤p < ∞. 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 ⊂Ω′, then DΩf(x)≤DΩ′f(x) and MΩf(x)≤MΩ′f(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 d≥1, Nagel, Stein and Wainger [20] showed that DΩis

bounded on all Lp(Rd+1 ), 1 < p ≤ ∞, when Ω = {(va1

i,...,vad+1

i) : i≥1}. Here

0< a1<··· < ad+1 are ﬁxed constants, and {vi:i≥1}is a sequence obeying

0< vi+1 ≤λvifor some lacunary constant 0 < λ < 1. Carbery [6] showed that DΩis

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 Ω = {(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 DΩon 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 DΩis 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 DΩand MΩin 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,d≥2 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 DΩand

MΩon 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 M≥3. 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 M−1.

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 M−2. 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|x−y| ≤ |γ(x)−γ(y)| ≤ C|x−y|,(1.5)

for some absolute constants 0< c < 1< C < ∞. Set Ω = {γ(t) : t∈ CM}. Then

5

(a) the set Ωadmits Kakeya-type sets,

(b) the operators DΩand MΩare unbounded on Lp(Rd+1)for all 1≤p < ∞.

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

minDΩf(x), MΩf(x)≥c0for x∈E∗

N(C0).(1.6)

This shows that

min||DΩ||p→p,||MΩ||p→p≥c0|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:N≥1}and {e

GN:N≥

1}are two collections of sets with |e

GN|/|GN| → ∞, enjoying the additional property

that for any point x∈e

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 DΩ. 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 DΩand MΩhas 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 M≥3 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 N≥1, typically much bigger than M. Let {Qt:t∈TN},

parametrized by the index set TN, be the collection of disjoint d-dimensional cubes

of sidelength M−Ngenerated by the lattice M−NZdin 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 N−1}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=d−dwill suﬃce.

Recall that the Nth iterate C[N]

Mof the Cantor construction is the union of 2Ndisjoint

intervals each of length M−N. 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)≤M−N

for every x∈ CM. Set

ΩN:= γ(D[N]

M),(2.2)

so that dist(ω, ΩN)≤CM −Nfor any ω∈Ω, with Cas in (1.5).

For any t∈TNand any ω∈ΩN, we deﬁne

Pt,ω := nr+sω :r∈e

Qt,0≤s≤10C0o,(2.3)

where C0is a large constant to be determined shortly (for instance, C0=ddc−1will

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·M−N. 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:TN→ΩNhas been speciﬁed, we set Pt,σ := Pt,σ(t). Thus

{Pt,σ :t∈TN}is a family of tubes rooted at the elements of an M−N-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) := [

t∈TN

Pt,σ.(2.4)

9

t1Pt1,σ

Pt2,σ

t2

Figure 1: Two typical tubes Pt1,σ and Pt2,σ rooted 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 N≥1, let ΣNbe a ﬁnite collection of slope assignments

from the lattice TNto the direction set ΩN. Every σ∈ΣNgenerates a set KN(σ)as

deﬁned in (2.4). Denote the power set of ΣNby P(ΣN).

Suppose that (ΣN,P(ΣN),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 C0≥1is a ﬁxed constant, and {aN},{bN}are deterministic sequences satisfying

aN

bN→ ∞,as N→ ∞.

Then Ωadmits Kakeya-type sets.

Proof. Fix any integer N≥1. 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|

4bN≤1

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| ≤ 4bN−1

≥3

4+3

4−1 = 1

2.

We may therefore choose a particular σ∈ΣNfor 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 ΩN⊂Ω 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, v2∈ΩNand t1, t2∈TN,t16=t2, let Pt1,v1and Pt2,v2be the tubes

deﬁned as in (2.3). If there exists p= (p1,···, pd+1)∈ Pt1,v1∩Pt2,v2, then the inequality

cen(Qt2)−cen(Qt1) + p1(v2−v1)≤2κd√dM−N,(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 x1∈e

Qt1,x2∈e

Qt2such that p=x1+p1v1=x2+p1v2, i.e., p1(v2−v1) = x1−x2.

The inequality (3.1) follows since |xi−cen(Qti)| ≤ κd√dM−Nfor 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(v2−v1)| ≥ κdM−N.(3.2)

Proof. Since t16=t2, we must have |cen(Qt1)−cen(Qt2)| ≥ M−N. Thus an intersection

is possible only if

p1|v2−v1| ≥ |cen(Qt2)−cen(Qt1)| − 2κd√dM−N≥(1 −2κd√d)M−N≥κdM−N,

where the ﬁrst inequality follows from (3.1) and the last inequality holds for an appro-

priate selection of κd.

Corollary 3.3. If t1∈TN,v1, v2∈ΩNand a cube Q⊆Rd+1 of sidelength C1M−Nwith

sides parallel to the coordinate axes are given, then there exists at most C2=C2(C1)

choices of t2∈TNsuch that Pt1,v1∩ Pt2,v2∩Q6=∅.

Proof. As p= (p1,···, pd+1) ranges in Q,p1ranges over an interval Iof length C1M−N.

If p∈ Pt1,v1∩Pt2,v2∩Q, the inequality (3.1) and the fact diam(Ω) ≤diam({1}×[−1,1]d) =

2√dimplies

cen(Qt2)−cen(Qt1) + cen(I)(v2−v1)≤ |(p1−cen(I))(v2−v1)|+ 2κd√dM−N

12

= 2√d(C1+κd)M−N,

restricting cen(Qt2) to lie in a cube of sidelength 2√d(C1+κd)M−Ncentred at cen(Qt1)−

cen(I)(v2−v1). 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:t∈TN,there exists v∈ΩNsuch that p∈ Pt,v.(3.3)

This set captures all the possible M−N-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:t∈TN, p ∈Pt,σ⊆Poss(p).

(b) For any p∈[0,10C0]×Rd,

Poss(p) = Qt:e

Qt∩(p−p1ΩN)6=∅(3.4)

⊆ {Qt:t∈TN, Qt∩(p−p1ΩN)6=∅}.(3.5)

Note that the set in (3.4) could be empty, but the one in (3.5) is not.

Proof. If p∈Pt,σ, then p∈ Pt,σ(t)with σ(t) equal to some v∈Ω. Thus Pt,v contains

pand hence Qt∈Poss(p), proving part (a). For part (b), we observe that p∈ Pt,v for

some v∈ΩNif and only if p−p1v∈e

Qt, i.e., e

Qt∩(p−p1ΩN)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, Ω2, on the set of directions given by the curve {(1, t, t2) : 0 ≤

t≤C}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, p−p1Ω2. The resulting Poss(p)set is thus

given by all cubes Qt,t∈TNsuch that e

Qtintersects a subset of the curve

{(0, p2−p1t, p3−p1t2) : 0 ≤t≤C}.

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−ℓ, M −ℓ+1)for some ℓ≤N−1. Then

#Qt:t∈TN, Qt∩Q6=∅,dist(Qt, p −p1ΩN)≤C1M−N≤C22N−ℓ.(3.6)

Proof. Let j∈Zbe the index such that M−j≤p1< 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 p−1

1M−N-separated points lying in

Q′:= x∈p−1

1Q: dist(x, p−1

1p−ΩN)≤C1p−1

1M−N.

But p−1

1p−ΩN= (1, c)−ΩNis an image of ΩNfollowing an inversion and translation.

This implies that there is a subset Ω′

Nof ΩN, depending on pand p−1

1Qand with diameter

O(Mj−ℓ), such that Q′is contained in a O(Mj−N)-neighborhood of −Ω′

N+ (1, c). The

number of Mj−N-separated points in Q′is comparable to that in Ω′

N.

14

Suppose ﬁrst that j≤ℓ. If C′⊆ C[N]

Mis deﬁned by the requirement Ω′

N=γ(C′),

then (1.5) implies that diam(C′) = O(Mj−ℓ). Thus C′is contained in at most O(1)

intervals of length Mj−ℓchosen at step ℓ−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 M−k−1. So the number of Mj−N-separated

points in C′, and hence γ(C′) is O(2(N−j)−(ℓ−j)) = O(2N−ℓ) as claimed. The case j≥ℓ

is even simpler, since the number of Mj−N-separated points in C′is trivially bounded

by 2N−j≤2N−ℓ.

Lemma 3.6. Fix t∈TNand p= (p1,···, pd+1)∈[M−ℓ, M −ℓ+1]×Rd, for some

0≤ℓ≪N. Let Qbe a cube centred at pof sidelength C1M−N. Then

#v∈ΩN:Q∩ Pt,v 6=∅≤C22ℓ.

Proof. If both Pt,v and Pt,v′have nonempty intersection with Q, then there exist q=

(q1,···, qd+1), q′= (q′

1,···, q′

d+1)∈Qsuch that both q−q1vand q′−q′

1v′land in e

Qt.

Thus,

p1|v−v′| ≤ |(q−p1v)−(q′−p1v′)|+|q−q′|

≤ |(q−q1v)−(q′−q′

1v′)|+|q1−p1||v|+|q′

1−p1||v′|+|q−q′|

≤(κd√d+ 10C1√d)M−N.

In other words, |v−v′| ≤ (10C1+κd)√dMℓ−N. 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 |α−α′| ≤

C2Mℓ−N. Thus there is a collection of at most O(1) chosen intervals at step N−ℓof

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(2ℓ).

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 C0≥1with the following properties.

(a) For any p∈[C0, C0+ 1] ×Rdand t∈TN, there exists at most one v∈ΩNsuch 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,Qt′∈Poss(p), let v=γ(α),v′=γ(α′)be the two

unique slopes in ΩNguaranteed by (a) such that p∈ Pt,v ∩ Pt′,v′. If kis the largest

integer such that Qtand Qt′are both contained in the same cube Q⊆ {0} × [0,1)d

of sidelength M−kwhose corners lie in M−kZd, then αand α′belong to the same

kth stage basic interval in the Cantor construction.

15

Proof. (a) Suppose v, v′∈ΩNare such that p∈ Pt,v ∩ Pt,v′. Then p−p1vand p−p1v′

both lie in e

Qt, so that p1|v−v′| ≤ κd√dM −N. Since p1≥C0and (1.5) holds, we

ﬁnd that

|α−α′| ≤ κd√d

cC0

M−N< M−N,

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 M−r. Therefore the inequality

above implies that both αand α′belong to the same basic Nth stage interval in

C[N]

M. But D[N]

Mcontains exactly one element from each such interval. So α=α′and

hence v=v′.

(b) If p∈ Pt,v ∩Pt′,v′, then p1|v−v′| ≤ diam( e

Qt∪e

Qt′)≤diam(Q) = √dM−k. Applying

(1.5) again combined with p1≥C0, we ﬁnd that |α−α′| ≤ √d

cC0M−k< M−k,for C0

chosen large enough. By the same property of the Cantor construction as used in

(a), we obtain that αand α′lie 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 ≤k≤n,hi1,...,iki ∈ T , where k= 0 corresponds to the empty

sequence, and

(ii.) for every j∈ {0,1,...,in}, we have hi1,...,in−1, ji ∈ T .

We say that hi1,...,in−1iis the parent of hi1,...,in−1, j iand that hi1,...,in−1, jiis

the (j+ 1)th child of hi1,...,in−1i. 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 u⊂v(see the remark

below). If u=hi1,...,imi ∈ T ,v=hj1,...,jni ∈ T ,m≤n, 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 u⊂vto 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 (in−1+

1)th child,..., of the (i1+ 1)th child of the root. All unordered pairs of the form

(hi1,...,in−1i,hi1,...,in−1, ini) describe the edges of the tree T. We say that the edge

originates at the vertex hi1,...,in−1iand that it terminates at the vertex hi1,...,in−1, 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 :k≥0}.

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 n∈Z+, we write the truncation of Tto its ﬁrst nlevels as

Tn={hi1,...,iki ∈ T : 0 ≤k≤n}.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

•u⊂vimplies 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 M≥2. We deﬁne an M-adic

rational as a number of the form i/Mkfor some i∈Z,k∈Z+, and an M-adic interval

as [i·M−k,(i+ 1) ·M−k). For any nonnegative integer iand positive integer ksuch that

i < Mk, there exists a unique representation

i=i1Mk−1+i2Mk−2+···+ik−1M+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 ij∈ZMfor each j. Naturally then, we deﬁne the tree of inﬁnite height

T([0,1); M) = {hi1,...,iki:k≥0, ij∈ZM}.(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·M−k,(i+ 1) ·M−k), 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 M−k. 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)·M−k,(ik(x) + 1) ·M−k). 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 x∈E. 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 M−k. More

precisely, hi1,...,iki ∈ Tk(E;M) if and only if E∩[i·M−k,(i+ 1) ·M−k)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 M≥3. 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 k≥1, consider the vertex hi1, i2,...,iki ∈ T . We know that ij∈ZM

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·M−k∈[0,1)d, we can

20

apply (4.2) to each component of ito obtain

i

Mk=i1

M+i2

M2+···+ik

Mk,

with ij∈Zd

Mfor all j. As before, we identify iwith hi1,...,iki.

Let φ:Zd

M→ {0,1,...,Md−1}be an enumeration of Zd

M. Deﬁne the full Md-adic

tree

T([0,1)d;M, φ) = nhφ(i1),...,φ(ik)i:k≥0,ij∈Zd

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:k≥0,ij∈Zd

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 M−k,

containing i·M−k, 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 ΩNbe the set deﬁned in (2.2).

(a) Given ΩN, there is a constant C1>0(depending only on dand C, c from (1.5))

such that for any 1≤k≤N, the number of kth generation vertices in TN(ΩN;M)

is ≤C12k.

(b) For any compact set K⊆Rd+1, there exists a constant C(K)>0with the following

property. For any x= (x1,···, xd+1)∈K, and 1≤k≤N, the number of kth

generation vertices in TN(E(x); M)is ≤C(K)2k, where E(x) := (x−x1ΩN)∩{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 CM−k. 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 Q∩ΩN6=∅. Then on one hand,

(x−x1Q)∩(x−x1ΩN)6=∅; on the other hand, the number of kth level M-adic cubes

covering (x−x1Q) 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 t⊆ustands 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

22

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 [

R∈∂T\

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

1−peY

∅⊂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.

23

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)

u∼v

+

−

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+R4−1

=(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</