VERTICAL PROJECTIONS IN THE HEISENBERG GROUP VIA CINEMATIC FUNCTIONS AND POINT-PLATE INCIDENCES KATRIN FÄSSLER AND TUOMAS ORPONEN

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VERTICAL PROJECTIONS IN THE HEISENBERG GROUP

VIA CINEMATIC FUNCTIONS AND POINT-PLATE INCIDENCES

KATRIN FÄSSLER AND TUOMAS ORPONEN

ABSTRACT. Let tπe:HÑWe:ePS1ube the family of vertical projections in the ﬁrst

Heisenberg group H. We prove that if KĂHis a Borel set with Hausdorff dimension

dimHKP r0,2s Y t3u, then

dimHπepKq ě dimHK

for H1almost every ePS1. This was known earlier if dimHKP r0,1s.

The proofs for dimHKP r0,2sand dimHK“3are based on different techniques.

For dimHKP r0,2s, we reduce matters to a Euclidean problem, and apply the method of

cinematic functions due to Pramanik, Yang, and Zahl.

To handle the case dimHK“3, we introduce a point-line duality between horizontal

lines and conical lines in R3. This allows us to transform the Heisenberg problem into a

point-plate incidence question in R3. To solve the latter, we apply a Kakeya inequality for

plates in R3, due to Guth, Wang, and Zhang. This method also yields partial results for

Borel sets KĂHwith dimHKP p5{2,3q.

CONTENTS

1. Introduction 2

1.1. Sharpness of the results 4

1.2. Proof outline for Theorem 1.7 4

Acknowledgements 5

2. Preliminaries on the Heisenberg group 5

3. Proof of Theorem 1.6 6

3.1. Projections induced by cinematic functions 6

3.2. From vertical projections to cinematic functions 8

4. Duality between horizontal lines and R311

4.1. Measures on the space of horizontal lines 13

4.2. Ball-plate duality 14

5. Discretising Theorem 1.7 18

6. Kakeya estimate of Guth, Wang, and Zhang 22

7. Proof of Theorem 1.7 28

Appendix A. Completing pδ, tq-sets to pδ, 3q-sets 30

References 31

Date: July 26, 2023.

2010 Mathematics Subject Classiﬁcation. 28A80 (primary) 28A78 (secondary).

Key words and phrases. Vertical projections, Heisenberg group, Hausdorff dimension, Incidences.

K.F. is supported by the Academy of Finland via the project Singular integrals, harmonic functions, and

boundary regularity in Heisenberg groups, grant No. 321696. T.O. is supported by the Academy of Finland via

the project Incidences on Fractals, grant No. 321896.

arXiv:2210.00458v3 [math.CA] 25 Jul 2023

2 KATRIN FÄSSLER AND TUOMAS ORPONEN

1. INTRODUCTION

Fix ePS1ˆ t0u Ă H, and consider the vertical plane We:“eKin the ﬁrst Heisenberg

group H, see Section 2for the deﬁnitions. Every point pPHcan be uniquely decomposed

as p“w¨v, where

wPWeand vPLe:“spanpeq.

This decomposition gives rise to the vertical projection πe:“πWe:HÑWe, deﬁned by

πeppq:“w. A good way to visualise πeis to note that the ﬁbres π´1

etwu,wPWe, coincide

with the horizontal lines w¨Le. These lines foliate H, as wranges in We, but are not

parallel. Thus, the projections πeare non-linear maps with linear ﬁbres. For example, in

the special cases e1“ p1,0,0qand e2“ p0,1,0qwe have the concrete formulae

πe1px, y, tq “ `0, y, t `xy

2˘and πe2px, y, tq “ `x, 0, t ´xy

2˘.(1.1)

From the point of view of geometric measure theory in the Heisenberg group, the vertical

projections are the Heisenberg analogues of orthogonal projections to pd´1q-planes in

Rd. One of the fundamental theorems concerning orthogonal projections in Rdis the

Marstrand-Mattila projection theorem [19,20]: if KĂRdis a Borel set, then

dimEπVpKq “ mintdimEK, d ´1u(1.2)

for almost all pd´1q-planes VĂRd. Here dimErefers to Hausdorff dimension in Eu-

clidean space – in contrast to the notation "dimH" which will refer to Hausdorff dimension

in the Heisenberg group. In Rd, orthogonal projections are Lipschitz maps, so the upper

bound in (1.2) is trivial, and the main interest in (1.2) is the lower bound.

The vertical projections πeare not Lipschitz maps HÑWerelative to the natural

metric dHin Hand We. Indeed, they can increase Hausdorff dimension: an easy ex-

ample is a horizontal line, which is 1-dimensional to begin with, but gets projected to a

2-dimensional set – a parabola – in almost all directions. For general (sharp) results on

how much πecan increase Hausdorff dimension, see [1, Theorem 1.3]. We note that the

vertical planes Wethemselves are 3-dimensional, and His 4-dimensional.

Can the vertical projections lower Hausdorff dimension? In some directions they can,

and the general (sharp) universal lower bound was already found in [1, Theorem 1.3]:

dimHπepKq ě maxt0,1

2pdimHK´1q,2 dimHK´5u, e PS1.

Our main result states that the dimension drop cannot occur in a set of directions of

positive measure for sets of dimension in r0,2sYt3u:

Theorem 1.3. Let KĂHbe a Borel set with dimHKP r0,2s Y t3u. Then dimHπepKq ě

dimHKfor H1almost every ePS1.

The result is sharp for all values dimHKP r0,2sYt3u, and new for dimHKP p1,2sYt3u.

It makes progress in [1, Conjecture 1.5] which proposes that

dimHπepKq ě mintdimHK, 3u(1.4)

for H1almost every ePS1. The cases dimHKP r0,1swere established around a decade

ago by Balogh, Durand-Cartagena, the ﬁrst author, Mattila, and Tyson [1, Theorem 1.4].

For dimHKą1, the strongest previous partial result is due to Harris [14] who in 2022

proved that

dimHπepKq ě min "1`dimHK

2,2*for H1a.e. ePS1.

VERTICAL PROJECTIONS 3

Other partial results, also higher dimensions, are contained in [2,4,13,15].

The "disconnected" assumption dimHKP r0,2s Y t3uis due to the fact that Theorem

1.3 is a combination of two separate results, with different proofs. Perhaps surprisingly,

the cases dimHKP r0,2sare a consequence of a "1-dimensional" projection theorem.

Namely, consider the (nonlinear) projections ρe:R3ÑRobtained as the t-coordinates of

the projections πe:

ρe“πT˝πe, πTpx, y, tq“p0,0, tq.(1.5)

Since the t-axis in His 2-dimensional, it is conceivable that the maps ρedo not a.e. lower

the Hausdorff dimension of Borel sets of dimension at most 2. This is what we prove:

Theorem 1.6. Let KĂR3be a Borel set. Then

dimEρepKq “ mintdimEK, 1uand dimHρepKq ě mintdimHK, 2u

for H1almost every ePS1. In fact, the following shaper conclusion holds: for 0ďsă

mintdimHK, 2u, we have dimEtePS1: dimHρepKq ď su ď s

Theorem 1.6 implies the cases dimHKP r0,2sof Theorem 1.3, because the map πTis

Lipschitz when restricted to any plane We, thus dimHπepKq ě dimHρepKqfor all ePS1.

The proof of Theorem 1.6 is a fairly straightforward application of recently developed

technology to study the restricted projections problem in R3(see [8,9,11,17,22]). Even

though the maps ρeare nonlinear, Theorem 1.6 falls within the scope of the cinematic func-

tion framework introduced by Pramanik, Yang, and Zahl [22]. In Theorem 3.2, we apply

this framework to record a more general version of Theorem 1.6 which simultaneously

generalises [22, Theorem 1.3] and Theorem 1.6. The details can be found in Section 3.

The case dimHK“3of Theorem 1.3 is the harder result. This time we do not know

how to deduce it from a purely Euclidean statement. Instead, it is deduced from the

following "mixed" result between Heisenberg and Euclidean metrics:

Theorem 1.7. Let KĂHbe a Borel set with dimHKě2. Then,

dimEπepKq ě mintdimHK´1,2u(1.8)

for H1almost every ePS1, and consequently

dimHπepKq ě mint2 dimHK´3,3u(1.9)

for H1almost every ePS1.

Theorem 1.7 will further be deduced from a δ-discretised result which may have inde-

pendent interest. We state here a simpliﬁed version (the full version is Theorem 5.11):

Theorem 1.10. Let 0ďtď3and ηą0. Then, the following holds for δ, ϵ ą0small enough,

depending only on η. Let Bbe a non-empty pδ, t, δ´ϵq-set of Heisenberg balls of radius δ, all

contained in BHp1q. Then, there exists ePS1such that

LebpπepYBqq ě δ3´t`η.(1.11)

Here Leb denotes Lebesgue measure on We, identiﬁed with R2. For the deﬁnition of

pδ, tq-sets of δ-balls, see Deﬁnition 5.1. Theorems 1.7 and 1.10 are proved in Sections 5-7.

Remark 1.12.It seems likely that the lower bound (1.11) remains valid under the alterna-

tive assumptions that |B| “ δ´tand

|tBPB:BĂBHpp, rqu| ď δ´ϵ¨´r

δ¯3

, p PH, r ěδ. (1.13)

4 KATRIN FÄSSLER AND TUOMAS ORPONEN

This is because the estimate (1.11) ultimately follows from Proposition 6.7 which works

under the non-concentration condition (1.13). We will not need this version of Theorem

1.10, so we omit the details.

1.1. Sharpness of the results. Theorem 1.3 is sharp for all values dimHKP r0,2s Y t3u.

The "mixed" inequality (1.8) in Theorem 1.7 is sharp for all values dimHKě2, even

though the Heisenberg corollary (1.9) is unlikely to be sharp for any value dimHKă3

(in fact, Theorem 1.3 shows that (1.9) is not sharp for dimHKă5{2q.

The sharpness examples are as follows: if s:“dimHKď2, take an s-dimensional

subset of the t-axis, and note that the t-axis is preserved by the projections ρeand πe.

If są2, take Kto be a union of translates of the t-axis, thus K:“K0ˆR. The πe-

projections send vertical lines to vertical lines, so πepKqis a union of vertical lines on We;

more precisely πepKq “ ¯πepK0q ˆ R, where ¯πeis an orthogonal projection in R2. These

observations lead to the sharpness of (1.8), and the sharpness of conjecture (1.4).

Theorem 1.10 is sharp for all values of tP r0,3s. Indeed, it is possible that |B| “

δ´t, and then LebpπepYBqq ≲δ3´tfor every ePS1. It also follows from (1.11) that

the smallest number of dH-balls of radius δneeded to cover πepYBqis ≳δ´t`η. One

might think that this solves Conjecture 1.4 for all dimHKP r0,3s, but we were not able

to make this deduction rigorous: the difﬁculty appears when attempting to δ-discretise

Conjecture 1.4, and is caused by the non-Lipschitz behaviour of πe:pH, dHqÑpWe, dHq.

This problem will be apparent in the proof of Theorem 1.7 in Section 7. Another, more

heuristic, way of understanding the difference between Theorem 1.10 and Conjecture 1.4

is this: LebpπepKqq is invariant under left-translating K, but dimHπepKqis generally not.

As we already explained, the proof of Theorem 1.6, therefore the cases dimHKP r0,2s

of Theorem 1.3, follow from recent developments in the theory of restricted projections in

R3, notably the cinematic function framework in [22]. The proof of Theorem 1.7 does not

directly overlap with these results (see Section 1.2 for more details), and for example does

not use the ℓ2-decoupling theorem, in contrast with [8,9,11]. That said, the argument

was certainly inspired by the recent developments in the restricted projection problem.

1.2. Proof outline for Theorem 1.7.The proof of Theorem 1.7 is mainly based on two

ingredients. The ﬁrst one is a point-line duality principle between horizontal lines in H,

and R3. To describe this principle, let LHbe the family of all horizontal lines in H, and let

LCbe the family of all lines in R3which are parallel to some line contained in a conical

surface C. In Section 4, we show that there exist maps ℓ:R3ÑLHand ℓ˚:HÑLC

(whose ranges cover almost all of LHand LC) which preserve incidence relations in the

following way:

qPℓppq ðñ pPℓ˚pqq, p PR3, q PH.

Thus, informally speaking, incidence-geometric questions between points in Hand lines

in LHcan always be transformed into incidence-geometric questions between points in

R3and lines in LC. The point-line duality principle described here was used implicitly

by Liu [18] to study Kakeya sets (formed by horizontal lines) in H. However, making

the principle explicit has already proved very useful since the ﬁrst version of this paper

appeared: we used it in [5] to study Kakeya sets associated with SLp2q-lines in R3, and

Harris [12] used it to treat the case dimHKą3of Theorem 1.3 (in this case the projections

πepKqturn out to have positive measure almost surely).

VERTICAL PROJECTIONS 5

The question about vertical projections in Hcan – after suitable discretisation – be

interpreted as an incidence geometric problem between points in Hand lines in LH. It

can therefore be transformed into an incidence-geometric problem between points in R3

and lines in LC. Which problem is this? It turns out that while the dual ℓ˚ppqof a point

pPHis a line in LC, the dual ℓ˚pBHqof a Heisenberg δ-ball resembles an δ-plate in R3–

a rectangle of dimensions 1ˆδˆδ2tangent to C. So, the task of proving Theorem 1.10

(hence Theorem 1.7) is (roughly) equivalent to the task of solving an incidence-geometric

problem between points in R3, and family of δ-plates.

Moreover: the plates in our problem appear as duals of certain Heisenberg δ-balls,

approximating a t-dimensional set KĂH, with 0ďtď3. Consequently, the plates

can be assumed to satisfy a t-dimensional "non-concentration condition" relative to the

metric dH. In common jargon, the plate family is a pδ, tq-set relative to dH.

In [10], Guth, Wang, and Zhang proved the sharp (reverse) square function estimate for the

cone in R3. A key component in their proof was a new incidence-geometric ("Kakeya")

estimate [10, Lemma 1.4] for points and δ-plates in R3(see Section 6for the details).

While this was not relevant in [10], it turns out that the incidence estimate in [10, Lemma

1.4] interacts perfectly with a pδ, 3q-set condition relative to dH. This allows us to prove,

roughly speaking, that the vertical projections of 3-Frostman measures on Hhave L2-

densities. See Corollary 5.6 for a more precise statement.

For 0ďtă3, the pδ, tq-set condition relative to dHno longer interacts so well with

[10, Lemma 1.4]. However, we were able to (roughly speaking) reduce Theorem 1.10 for

pδ, tq-sets, 0ďtď3, to the special case t“3. This argument is explained in Section 5, so

we omit the discussion here.

Acknowledgements. We thank the reviewer for a careful reading of the manuscript, and

for providing us with helpful comments.

2. PRELIMINARIES ON THE HEISENBERG GROUP

We brieﬂy introduce the Heisenberg group and relevant related concepts. A more

thorough introduction to the geometry of the Heisenberg group can be found in many

places, for instance in the monograph [3].

The Heisenberg group H“ pR3,¨q is the set R3equipped with the non-commutative

group product deﬁned by

px, y, tq¨px1, y1, t1q “ `x`x1, y `y1, t `t1`1

2pxy1´yx1q˘.

The Heisenberg dilations are the group automorphisms δλ,λą0, deﬁned by

δλpx, y, tq“pλx, λy, λ2tq.

The group product gives rise to projection-type mappings onto subgroups that are in-

variant under Heisenberg dilations. For ePS1, we deﬁne the horizontal subgroup

Le:“ tpse, 0q:sPRu.

The vertical subgroup Weis the Euclidean orthogonal complement of Lein R3; in particu-

lar it is a plane containing the vertical axis. Every point pPHcan be written in a unique

way as a product p“pWe¨pLewith pWePWeand pLePLe. The vertical Heisenberg

projection onto the vertical plane Weis the map

πe:HÑWe, p “pWe¨pLeÞÑ pWe.

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VERTICALPROJECTIONSINTHEHEISENBERGGROUPVIACINEMATICFUNCTIONSANDPOINT-PLATEINCIDENCESKATRINFÄSSLERANDTUOMASORPONENABSTRACT.Lettπe:HÑWe:ePS1ubethefamilyofverticalprojectionsinthefirstHeisenberggroupH.WeprovethatifKĂHisaBorelsetwithHausdorffdimensiondimHKPr0,2sYt3u,thendimHπepKqědimHKforH1almosteveryeP...

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VERTICAL PROJECTIONS IN THE HEISENBERG GROUP VIA CINEMATIC FUNCTIONS AND POINT-PLATE INCIDENCES KATRIN FÄSSLER AND TUOMAS ORPONEN

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