Vertical projections in the Heisenberg group
via cinematic functions and point-plate incidences
Abstract.
Let be the family of vertical projections in the first Heisenberg group . We prove that if is a Borel set with Hausdorff dimension , then
for almost every . This was known earlier if .
The proofs for and are based on different techniques. For , we reduce matters to a Euclidean problem, and apply the method of cinematic functions due to Pramanik, Yang, and Zahl.
To handle the case , we introduce a point-line duality between horizontal lines and conical lines in . This allows us to transform the Heisenberg problem into a point-plate incidence question in . To solve the latter, we apply a Kakeya inequality for plates in , due to Guth, Wang, and Zhang. This method also yields partial results for Borel sets with .
Key words and phrases:
Vertical projections, Heisenberg group, Hausdorff dimension, Incidences2010 Mathematics Subject Classification:
28A80 (primary) 28A78 (secondary)1. Introduction
Fix , and consider the vertical plane in the first Heisenberg group , see Section 2 for the definitions. Every point can be uniquely decomposed as , where
This decomposition gives rise to the vertical projection , defined by . A good way to visualise is to note that the fibres , , coincide with the horizontal lines . These lines foliate , as ranges in , but are not parallel. Thus, the projections are non-linear maps with linear fibres. For example, in the special cases and we have the concrete formulae
| (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 -planes in . One of the fundamental theorems concerning orthogonal projections in is the Marstrand-Mattila projection theorem [19, 20]: if is a Borel set, then
| (1.2) |
for almost all -planes . Here refers to Hausdorff dimension in Euclidean space – in contrast to the notation "" which will refer to Hausdorff dimension in the Heisenberg group. In , 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 are not Lipschitz maps relative to the natural metric in and . Indeed, they can increase Hausdorff dimension: an easy example is a horizontal line, which is -dimensional to begin with, but gets projected to a -dimensional set – a parabola – in almost all directions. For general (sharp) results on how much can increase Hausdorff dimension, see [1, Theorem 1.3]. We note that the vertical planes themselves are -dimensional, and is -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]:
Our main result states that the dimension drop cannot occur in a set of directions of positive measure for sets of dimension in :
Theorem 1.3.
Let be a Borel set with . Then for almost every .
The result is sharp for all values , and new for . It makes progress in [1, Conjecture 1.5] which proposes that
| (1.4) |
for almost every . The cases were established around a decade ago by Balogh, Durand-Cartagena, the first author, Mattila, and Tyson [1, Theorem 1.4]. For , the strongest previous partial result is due to Harris [14] who in 2022 proved that
Other partial results, also higher dimensions, are contained in [2, 4, 13, 15].
The "disconnected" assumption is due to the fact that Theorem 1.3 is a combination of two separate results, with different proofs. Perhaps surprisingly, the cases are a consequence of a "-dimensional" projection theorem. Namely, consider the (nonlinear) projections obtained as the -coordinates of the projections :
| (1.5) |
Since the -axis in is -dimensional, it is conceivable that the maps do not a.e. lower the Hausdorff dimension of Borel sets of dimension at most . This is what we prove:
Theorem 1.6.
Let be a Borel set. Then
for almost every . In fact, the following shaper conclusion holds: for , we have .
Theorem 1.6 implies the cases of Theorem 1.3, because the map is Lipschitz when restricted to any plane , thus for all .
The proof of Theorem 1.6 is a fairly straightforward application of recently developed technology to study the restricted projections problem in (see [8, 9, 11, 17, 22]). Even though the maps are nonlinear, Theorem 1.6 falls within the scope of the cinematic function 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 of 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 be a Borel set with . Then,
| (1.8) |
for almost every , and consequently
| (1.9) |
for almost every .
Theorem 1.7 will further be deduced from a -discretised result which may have independent interest. We state here a simplified version (the full version is Theorem 5.11):
Theorem 1.10.
Let and . Then, the following holds for small enough, depending only on . Let be a non-empty -set of Heisenberg balls of radius , all contained in . Then, there exists such that
| (1.11) |
Here denotes Lebesgue measure on , identified with . For the definition of -sets of -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 alternative assumptions that and
| (1.13) |
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 . The "mixed" inequality (1.8) in Theorem 1.7 is sharp for all values , even though the Heisenberg corollary (1.9) is unlikely to be sharp for any value (in fact, Theorem 1.3 shows that (1.9) is not sharp for .
The sharpness examples are as follows: if , take an -dimensional subset of the -axis, and note that the -axis is preserved by the projections and . If , take to be a union of translates of the -axis, thus . The -projections send vertical lines to vertical lines, so is a union of vertical lines on ; more precisely , where is an orthogonal projection in . 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 . Indeed, it is possible that , and then for every . It also follows from (1.11) that the smallest number of -balls of radius needed to cover is . One might think that this solves Conjecture 1.4 for all , but we were not able to make this deduction rigorous: the difficulty appears when attempting to -discretise Conjecture 1.4, and is caused by the non-Lipschitz behaviour of . 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: is invariant under left-translating , but is generally not.
As we already explained, the proof of Theorem 1.6, therefore the cases of Theorem 1.3, follow from recent developments in the theory of restricted projections in , 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 -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 , and . To describe this principle, let be the family of all horizontal lines in , and let be the family of all lines in which are parallel to some line contained in a conical surface . In Section 4, we show that there exist maps and (whose ranges cover almost all of and ) which preserve incidence relations in the following way:
Thus, informally speaking, incidence-geometric questions between points in and lines in can always be transformed into incidence-geometric questions between points in and lines in . The point-line duality principle described here was used implicitly by Liu [18] to study Kakeya sets (formed by horizontal lines) in . 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 -lines in , and Harris [12] used it to treat the case of Theorem 1.3 (in this case the projections turn out to have positive measure almost surely).
The question about vertical projections in can – after suitable discretisation – be interpreted as an incidence geometric problem between points in and lines in . It can therefore be transformed into an incidence-geometric problem between points in and lines in . Which problem is this? It turns out that while the dual of a point is a line in , the dual of a Heisenberg -ball resembles an -plate in – a rectangle of dimensions tangent to . 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 , and family of -plates.
Moreover: the plates in our problem appear as duals of certain Heisenberg -balls, approximating a -dimensional set , with . Consequently, the plates can be assumed to satisfy a -dimensional "non-concentration condition" relative to the metric . In common jargon, the plate family is a -set relative to .
In [10], Guth, Wang, and Zhang proved the sharp (reverse) square function estimate for the cone in . A key component in their proof was a new incidence-geometric ("Kakeya") estimate [10, Lemma 1.4] for points and -plates in (see Section 6 for 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 -set condition relative to . This allows us to prove, roughly speaking, that the vertical projections of -Frostman measures on have -densities. See Corollary 5.6 for a more precise statement.
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 is the set equipped with the non-commutative group product defined by
The Heisenberg dilations are the group automorphisms , , defined by
The group product gives rise to projection-type mappings onto subgroups that are invariant under Heisenberg dilations. For , we define the horizontal subgroup
The vertical subgroup is the Euclidean orthogonal complement of in ; in particular it is a plane containing the vertical axis. Every point can be written in a unique way as a product with and . The vertical Heisenberg projection onto the vertical plane is the map
The vertical projection to the -plane will play a special role; this projection will be denoted , and it has the explicit formula stated in (1.1). Preliminaries about Heisenberg projections can be found for instance in [21, 2, 1]. These mappings have turned out to play an important role in geometric measure theory of the Heisenberg group endowed with a left-invariant non-Euclidean metric. The Korányi metric is defined by
where is the Korányi norm given by
We will use the symbol to denote the ball centered at with radius with respect to the Korányi metric. Balls centred at the origin are denoted . All vertical planes , , equipped with are isometric to each other via rotations of about the vertical axis. The Heisenberg dilations are similarities with respect to , and it is easy to see that is a -regular space, while the vertical subgroups are -regular with respect to . Moreover, there exists a constant , independent of , such that under the obvious identification of with , the restriction of the -dimensional Hausdorff measure to agrees with the -dimensional Lebesgue measure on up to the multiplicative constant .
Vertical projections are neither group homomorphisms nor Lipschitz mappings with respect . However, they behave well with respect to the Lebesgue measure on vertical planes. Namely, for every Borel set , we have that
| (2.1) |
see the formula at the bottom of page 1970 in the proof of [7, Lemma 2.20].
3. Proof of Theorem 1.6
In this section, we prove Theorem 1.6, and therefore the cases of Theorem 1.3. Further, Theorem 1.6 will be inferred from a more general statement, Theorem 3.2, modelled after [22, Theorem 1.3]. We first discuss Theorem 3.2, and then explain in Section 3.2 how it can be applied to deduce Theorem 1.6.
3.1. Projections induced by cinematic functions
We start by introducing terminology from [22, Definition 1.6] which will be needed for the formulation of Theorem 3.2.
Definition 3.1 (Cinematic family).
Let be a compact interval, and let be a family of functions satisfying the following conditions:
-
(1)
is a compact interval, and has finite diameter in .
-
(2)
is a doubling metric space.
-
(3)
For all , we have
Then, is called a cinematic family.
The following projection theorem is modelled after [22, Theorem 1.3]:
Theorem 3.2.
Let , let be a compact interval, and let be a family of -Lipschitz maps , where is a ball. For , define the function by . Assume that is a bilipschitz embedding , and assume that is a cinematic family.
Then, the projections satisfy (3.8): if is a Borel set, then
We only sketch the proof of Theorem 3.2 since it is virtually the same as the proof of [22, Theorem 1.3]: this is the special case of Theorem 3.2, where
| (3.3) |
and parametrises a curve on satisfying (this condition is needed to guarantee that the family is cinematic for every ball , see the proof of [22, Proposition 2.1]).
The proof of [22, Theorem 1.3] is based on a reduction to [22, Theorem 1.7]. This is a "Kakeya-type" estimate concerning -neighbourhoods of graphs of cinematic functions. More precisely, [22, Theorem 1.7] is only used via [22, Proposition 2.1], a special case of [22, Theorem 1.7] concerning the cinematic family . We formulate a more general version of this proposition below: the only difference is that the cinematic family is replaced by the family relevant for Theorem 3.2:
Proposition 3.4.
Fix and . Let be a compact interval, let be a ball, and let be a family of uniformly Lipschitz functions with the properties assumed in Theorem 3.2: thus, is a cinematic family, and the map is a bilipschitz embedding , where . Then there exists such that the following holds for all :
Let be a quasi-product. Let be a -separated set that satisfies
| (3.5) |
Then
where is absolute, and is the -neighbourhood of the graph of .
Proof.
The proof of [22, Proposition 2.1] is easy (given [22, Theorem 1.7]), but the proof of Proposition 3.4 is almost trivial. Indeed, the first part in the proof of [22, Proposition 2.1] is to verify that the family is cinematic in the case , but this is already a part of our hypothesis. The second part in the proof of [22, Proposition 2.1] is to verify that is a bilipschitz embedding , and this is – again – part of our hypothesis. In other words, all the work in the proof of [22, Proposition 2.1] has been made part of the hypotheses of Proposition 3.4. ∎
The reduction from [22, Theorem 1.3] to [22, Proposition 2.1] (in our case from Theorem 3.2 to Proposition 3.4) is presented in [22, Sections 2.1-2.4], and does not use the special form (3.3) (for example the linearity) of the maps : it is only needed that
-
(1)
the maps are uniformly Lipschitz, for ,
-
(2)
.
Property (1) is assumed in Theorem 3.2, whereas property (2) follows from the assumption that the family is cinematic (and in particular a bounded subset of ).
The argument in [22, Sections 2.1-2.4] is extremely well-written, and our notation is deliberately the same, so we will not copy the whole proof. We only make a few remarks, below. If the reader is unfamiliar with the ideas involved, we warmly recommend reading first the heuristic section [22, Section 1.2].
Proof sketch of Theorem 3.2.
The argument in [22, Section 2.1] can be copied verbatim; nothing changes. The most substantial change occurs in [22, Section 2.2]. Namely, [22, (2.10)] uses the fact (true in [22]) that the -image of a -cube has length . For the general Lipschitz maps in Theorem 3.2 this may not be the case; it would be true for the special maps needed in Theorem 3.7, so also this part of [22] would work verbatim for these maps. However, even in the generality of Theorem 3.2 the problem can be completely removed: one only needs to replace every occurrence of in [22, Section 2.2] by an interval
of length centred at , where is the centre of . Since only appears as a "tool" in [22, Section 2.2], the rest of the argument will remain unchanged. Let us, however, discuss what changes in [22, Section 2.2] when is replaced by . We assume familiarity with the notation in [22].
First and foremost, [22, (2.9)] remains valid: whenever is a cube that intersects , then by our assumption that the maps are -Lipschitz. Therefore,
This gives [22, (2.9)] with the slightly modified definition of , stated above. Consequently, also the version of [22, (2.10)] is true where is replaced by : here the length bound is used. Finally, to deduce [22, (2.13)] from [22, (2.10)], we need to know that [22, (2.12)] remains valid when is replaced with . This is clear: if , then by definition, and therefore , where
is the analogue of [22, (1.12)], and is the -neighbourhood of . We have now verified [22, (2.13)]. The intervals or play no further role in the proof. The rest of [22, Section 2.2] works verbatim.
3.2. From vertical projections to cinematic functions
We explain how the general projection result, Theorem 3.2, can be applied to prove Theorem 3.7, which concerns the special projections . Recall that is the vertical projection to the plane . For , we write is the counterclockwise rotation of by . With this notation, the map has the explicit formula
| (3.6) |
where is the Euclidean dot product in . In the formula (3.6), we have also identified each plane with via the map . It is worth noting that the distance restricted to the plane (for fixed) is bilipschitz equivalent to the parabolic distance on , namely .
With the explicit expression (3.6) in hand, the nonlinear projections introduced in (1.5) have the following formula:
By a slight abuse of notation, we write "" for the square root metric on : thus . The projection restricted to any fixed plane is a Lipschitz map , even though is not "globally" a Lipschitz map . Therefore for all , and the cases of Theorem 1.3 follow from Theorem 1.6, whose contents are repeated here:
Theorem 3.7.
Let be Borel, and let . Then,
| (3.8) |
As a consequence, for every ,
| (3.9) |
In particular, for almost every .
Remark 3.10.
We explain why (3.8) implies (3.9). It is well-known that
for all sets . This simply follows from the fact that the identity map is locally -Hölder continuous. Therefore, if , as in (3.9), we have , and (3.8) is applicable. Since
(the square root metric on doubles Euclidean dimension), we have
This is what we claimed in (3.9).
For the remainder of this section, we focus on proving the Euclidean statement (3.8). This is chiefly based on verifying that the projections give rise to a cinematic family of functions, as in Definition 3.1. Let us introduce the relevant cinematic family. We re-parametrise the projections , , as , , where
With this notation, we define the following functions , :
| (3.11) |
Proposition 3.12.
Let . Then, there exists a radius such that is a cinematic family.
The compact interval appearing in conditions (1)-(3) of Definition 3.1 can be taken to be – this makes no difference, since the functions are -periodic. It turns out that the conditions (1)-(2) are satisfied for the family , whenever is an arbitrary ball. To verify condition (3), we will need to assume that lies outside the -axis; we will return to this a little later. We first compute the derivatives of the functions in . For , we have
This expression can be further simplified by noting that , and . Therefore,
| (3.13) |
From this expression, we may compute the second derivative:
| (3.14) |
The formulae (3.11)-(3.14) immediately show that the map is locally Lipschitz:
| (3.15) |
This implies conditions (1)-(2) in Definition 3.1 for the family . Regarding condition (3) in Definition 3.1, we claim the following:
Proposition 3.16.
If , there exists a radius and a constant such that
| (3.17) |
for all and .
We start with the following lemma:
Lemma 3.18.
For every there exists a constant and a radius such that the following holds:
| (3.19) |
Proof.
Recall that and . We then define by
Then, we note that , so in particular the Jacobian of is non-vanishing outside the -axis. Now (3.19) follows from the inverse function theorem. ∎
We then prove Proposition 3.16:
Proof of Proposition 3.16.
To deduce (3.17) from (3.19), we record the following rotation invariance:
| (3.20) |
Here is a counterclockwise rotation around the -axis. The proof is evident from the formulae (3.11)-(3.14), and noting that
Now we are in a position to conclude the proof of (3.17). Fix and . Then, apply Lemma 3.18 to the point
This yields a constant and a radius such that
| (3.21) |
for all . Next, we choose to be a sufficiently short interval around such that the following holds:
Then, it follows from a combination of (3.20) and (3.21) that
for all and all . This completes the proof of (3.17) of all . To extend the argument of all , note that the functions , and all of their derivatives, are -periodic. So, it suffices to show that (3.17) holds for . This follows by compactness from what we have already proven, by covering by finitely many intervals of the form , and finally defining "" and "" to be the minima of the constants and obtained in the process. ∎
Proposition 3.12 now follows from Proposition 3.16, and the discussion above it (where we verified Definition 3.1(1)-(2)). We then conclude the proof of Theorem 3.7:
Proof of Theorem 3.7.
Given Remark 3.10, it suffices to prove (3.8), which will be a consequence of Theorem 3.2. Indeed, since the projections are isometries on the -axis, we may assume that
Consequently, for , we may fix a point outside the -axis such that
| (3.22) |
Apply Proposition 3.12 to find a radius such that the family of functions is cinematic. It follows from a combination of (3.15) and Proposition 3.16 that is a bilipschitz embedding . Therefore Theorem 3.2 is applicable: for every we have
4. Duality between horizontal lines and
This section contains preliminaries to prove Theorem 1.7. Most importantly, we introduce a notion of duality that associates to points and horizontal lines in certain lines and points in . The lines in will be light rays – translates of lines on a fixed conical surface. To define these, we let be the vertical cone
and we denote by the () rotated cone
where . The cone is foliated by lines
| (4.1) |
cf. the proof of [18, Theorem 1.2], where a similar parametrization is used. To be accurate, the lines only foliate . We will abuse notation by writing for the parametrisation of the line .
Definition 4.2 (Light rays).
We say that a line in is a light ray if for some and . In other words, is a (Euclidean) translate of a line contained in (excluding the -axis).
Remark 4.3.
Every light ray can be written as for a unique .
Definition 4.4 (Horizontal lines).
A line in is horizontal if it is a Heisenberg left translate of a horizontal subgroup, that is, there exists and such that .
Remark 4.5.
Every horizontal line, apart from left translates of the -axis, can be written as for a uniquely determined point .
Definition 4.6.
We define the following correspondence between points and lines:
- •
-
•
To a point , we associate the horizontal line
Given a set of points in , we define the family of light rays
| (4.8) |
Remark 4.9.
It is worth observing that the point appearing in formula (4.7) is nearly the vertical projection of to the -plane; the actual formula for this projection would be . It follows from this observation that
| (4.10) |
because .
Under the point-line correspondence in Definition 4.6, incidences between points and horizontal lines in are in one-to-one correspondence with incidences between light rays and points in .
Lemma 4.11 (Incidences are preserved under duality).
For and , we have
Proof.
Let and . The condition is equivalent to
Recalling the notation , this is further equivalent to
| (4.12) |
Finally, (4.12) is equivalent to . ∎
4.1. Measures on the space of horizontal lines
The duality between points in and horizontal lines in Definition 4.6 allows one to push-forward Lebesgue measure "" on to construct a measure "" on the set of horizontal lines:
There is, however, a more commonly used measure on the space of horizontal lines. This measure "" is discussed extensively for example in [6, Section 2.3]. The measure is the unique (up to a multiplicative constant) non-zero left invariant measure on the set of horizontal lines. One possible formula for it is the following:
| (4.13) |
Let , and consider the weighted measure . Then, starting from the definition (4.13), it is easy to check that
| (4.14) |
where .
While the measure is mutually absolutely continuous with respect to , the Radon-Nikodym derivative is not bounded (from above and below): with our current notational conventions, the lines are never parallel to the -axis, and the -density of lines making a small angle with the -axis is smaller than their -density. The problem can be removed by restricting our considerations to lines which make a substantial angle with the -axis. For example, let be the set of horizontal lines which have slope at most relative to the -axis; thus
Then, for all Borel sets . The lines in coincide with pre-images of the form , , where consists of those vectors making an angle at most with the -axis. Now, (4.14) also holds in the following restricted form:
| (4.15) |
This equation will be useful in establishing Theorem 5.2. This will, formally, only prove Theorem 5.2 with "" in place of "", but the original version is easy to deduce from this apparently weaker version.
4.2. Ball-plate duality
Recall from (4.8) the definition of the (dual) line set for . What does look like? The answer is: a plate tangent to the cone . Informally speaking, for , an -plate tangent to is a rectangle of dimensions whose long side is parallel to a light ray, and whose orientation is such that the plate is roughly tangent to , see Figure 1. To prove rigorously that looks like such a plate (inside ), we need to be more precise with the definitions.
Recall that the cone is a rotation of the "standard" cone .
The intersection of with the plane is the parabola
For every and , choose a rectangle of dimensions in the plane , centred at , such that the longer -side is parallel to the tangent line of at . Then for an absolute constant . Now, the -plate centred at is the set obtained by sliding the rectangle along the light ray containing inside , see Figure 1. We make this even more formal in the next definition.
Definition 4.16 (-plate).
Let , and let with . Let , and define , where
(The rectangle is the intersection of an -plate with the plane .) Define
The set is called the -plate centred at . In general, an -plate is any translate of one of the sets , for with , and .
For the -plate , we also commonly use the notation , where .
Remark 4.17.
Since we require in Definition 4.16, it is clear that an -plate contains, and is contained in, a rectangle of dimensions . It is instructive to note that the number of "essentially distinct" -plates intersecting is roughly : to see this, take a maximal -separated subset of , and note that for each , the plate has volume . Therefore it takes translates of to cover . This -numerology already suggests that the various -plates might correspond to Heisenberg -balls via duality.
To relate the plates to Heisenberg balls, we define a slight modification of the plates . Whereas is a union of (truncated) light rays in one fixed direction, the following "modified" plates contain full light rays in an -arc of directions. These "modified" plates will finally match the duals of Heisenberg balls, see Proposition 4.22.
Definition 4.18 (Modified -plate).
Let and . Let be the rectangle from Definition 4.16. For , define the modified -plate
| (4.19) |
Remark 4.20.
The relation between the sets and is that the following holds for some absolute constant : if , , and , then
| (4.21) |
(The constant "" is arbitrary, but happens to be the one we need.) To see this, it suffices to check the case . Consider the "slices" of and with a fixed plane for . If , both slices coincide with the rectangle . If , the slice can be written as a sum
whereas . The relationship between these two slices is depicted in Figure 2.
After this, we leave it to the reader to verify that if is sufficiently small, and for .
We record the following consequence of (4.21): is contained in a tube of width around the line . This is because is obviously contained in a tube of width around (this is a very non-sharp statement, using only that the longer side of has length .)
We then show that the -duals of Heisenberg balls are essentially modified plates:
Proposition 4.22.
Let , , and . Then,
| (4.23) |
where is an absolute constant, and .
Remark 4.24.
To build a geometric intuition, it will be helpful to notice the following. The -coordinate of the point is "". On the other hand, while the modified plate contains many lines, they are all "close" to the "central" line (see Definition 4.19). According to the inclusions in (4.23), this means that the "direction" of the modified plate containing the dual is determined by the -coordinate of . Even less formally: Heisenberg balls whose centres have the same -coordinate are dual to parallel plates.
Proof of Proposition 4.22.
To prove the inclusion , let , and write with and . First, we note that
| (4.25) |
Let be the vertical projection to the -plane . Then by the definition of . We now observe that , so
We claim that
| (4.26) |
This will prove that
| (4.27) |
Recalling the definition (4.19), a combination of (4.25) and (4.27) now shows that
This will complete the proof of the inclusion .
Let us then prove (4.26). Pick . Then,
so
| (4.28) |
Now, to prove (4.26), recall that . Thus, we need to show that . Equivalently, . Recalling the definition of , one checks that
Using (4.28), we finally note that the point on the right lies in the parabolic rectangle . This concludes the proof of (4.26).
Let us then prove the inclusion . The set is a union of the lines , where and . We need to show that every such line can be realised as for some . In this task, we are aided by the formula
observed in (4.7). This formula shows that we need to define , where , and is as in "". Then we just have to hope that .
Recalling that , one can check by direct computation that
| (4.29) |
On the other hand, one may easily check that is equivalent to
which implies and . Since moreover by assumption, it follows from (4.29) and the definition of the norm that . This completes the proof. ∎
We close the section with two additional auxiliary results:
Proposition 4.30.
Let and , and assume that . Assume moreover that . Then for some absolute constant .
Proof.
Write , so that . Since , in particular . By the previous proposition, we already know that
where we have written . Since , we know that . But
so we may deduce that
| (4.31) |
Moreover, in Remark 4.20 we noted that is contained in the -neighbourhood of the line . Therefore also . This implies that , and hence .
We already noted in Remark 4.24 that the (modified) -plates containing and have (almost) the same direction if the points have (almost) the same -coordinate. In this case, if , it is natural to expect that and are disjoint, at least inside . The next lemma verifies this intuition.
Lemma 4.33.
Let and be points with the properties
| (4.34) |
Then, .
Proof.
We may reduce to the case by the following argument. Start by choosing a point such that the -coordinate of equals . This is possible, because , and the projection of to the -plane is a Euclidean disc of radius . Then, notice that , so
Now, if we have already proven the lemma in the case (and for "" in place of ""), it follows that , and finally .
Let us then assume that . It follows from (4.34) and the first inclusion in Proposition 4.22 combined with the first inclusion in (4.21) that
for some absolute constant . Let "" be a point in the intersection, and (using the definition of ), express in the two following ways:
where and , and . The terms conveniently cancel out, and we find that
or equivalently
| (4.35) |
We have already computed in (4.32) that
and now it follows immediately from (4.35) that . ∎
5. Discretising Theorem 1.7
The purpose of this section is to reduce the proof of Theorem 1.7 to Theorem 5.2 which concerns -sets. We start by defining these precisely:
Definition 5.1 ((-set).
Let be a metric space, and let and . A non-empty bounded set is called a -set if
Here is the smallest number of balls of radius needed to cover . A family of sets (typically: disjoint -balls) is called a -set if is a -set.
If , or , the -set condition is always tested relative to the metric . We then state a -discretised version of Theorem 1.7 for sets of dimension :
Theorem 5.2.
For every , there exists and such that the following holds for all . Let be a non-empty -set of -balls contained in , with -separated centres. Let be the measure on with density
| (5.3) |
Then,
The proof of Theorem 5.2 will be given in Section 6. Deducing Theorem 1.7 from Theorem 5.2 involves two steps. The first one, carried out in Section 7, is to reduce Theorem 1.7 to a -discretised version, which concerns -sets with all possible values . This statement is Theorem 5.11 below, a simplified version of which was stated as Theorem 1.10 in the introduction.
The second – and less standard – step, carried out in this section, is to deduce Theorem 5.11 from Theorem 5.2. Heuristically, Theorem 5.2 is nothing but the -dimensional case of Theorem 5.11 – although in this case the statement looks more quantitative. We therefore need to argue that if we already have Theorem 5.11 for sets of dimension , then we also have it for sets of dimension . The heuristic is simple: given a set of dimension , we start by "adding" (from the left) to another – random – set of dimension . Then, we apply the -dimensional version of Theorem 5.11 to , and this gives the correct conclusion for . A crucial point is that Theorem 5.11 concerns the Lebesgue measure (not the dimension) of . This quantity is invariant under left translating . This allows us to control in a useful way.
We turn to the details. To deduce Theorem 1.7 from Theorem 5.2, we need a corollary of Theorem 5.2, stated in Corollary 5.6, which concerns slightly more general measures than ones of the form (as in (5.3)):
Definition 5.4 (-measure).
Let and . A Borel measure on is called a -measure if has a density with respect to Lebesgue measure, also denoted , and the density satisfies
If the constant irrelevant, a -measure may also be called a -measure.
We will use the following notion of -truncated Riesz energy:
| (5.5) |
where is a Radon measure, , and .
Corollary 5.6.
For every , there exists such that the following holds for all and . Let be a -probability measure on with . Then, there exists a Borel set such that , and
| (5.7) |
Proof.
Fix , , and . The dependence of on will eventually be determined by an application of Theorem 5.2, but we will require at least that .
It follows from and Chebychev’s inequality that there exists a set of measure such that for all and . Now, for dyadic rationals , let
We discard immediately the sets with : the union of these sets has measure for small enough, so , where
Now, is covered by the sets with , and the number of such sets is . We let be an enumeration of these values of "", and we abbreviate . We note that the union of the sets with has measure at most (for small), so finally
has measure . Moreover, is covered by the sets with . Re-indexing if necessary, we now assume that for all .
For fixed, let be a finitely overlapping (Vitali) cover of by balls of radius , centred at . Using the facts and , and the uniform lower bound for , it is easy to check that each is a -set with
| (5.8) |
Thus, writing
and assuming that are sufficiently small in terms of , we may deduce from Theorem 5.2 that
Finally, it follows from the -property of that
Thus, also the density of is bounded from above by the density of :
This completes the proof of (5.7) (with "" in place of ""). ∎
The concrete -measures we will consider have the form , where has a density of the form (5.3) (these are almost trivially -measures), and is a (discrete) probability measure. The notation refers to the (non-commutative!) Heisenberg convolution of and , that is, the push-forward of under the group product . Let us verify that such measures are also -measures:
Lemma 5.9.
Let be measure, and let be an arbitrary Borel probability measure on . Then is again a -measure.
Proof.
Recall that a measure is absolutely continuous by definition, so the notation "" is well-defined for Lebesgue almost every . The following formulae are valid, and easy to check, for Lebesgue almost every :
and
| (5.10) |
Now, if one applies the -measure assumption to the formula on the left hand side, one obtains
Lebesgue measure is invariant under left translations, so
Therefore, it follows from equation (5.10) that
for Lebesgue almost every . This is what we claimed. ∎
We are then ready to state and prove the -discretised counterpart of Theorem 1.7.
Theorem 5.11.
Let . Then, there exist , depending only on , such that the following holds for all . Let be a set of -balls with -separated centres, all contained in , and let be a Borel set of length . Then, there exists such that the following holds: if is any sub-family with , then
In particular, cannot be covered by fewer than parabolic balls of radius .
Proof.
To reach a contradiction, assume that there exists a -set of -balls with -separated centres, contained in , and violating the conclusion of Theorem 5.11: there exists , and for every (Borel subset of of length ), there exists a subset with with the property
| (5.12) |
We aim for a contradiction if are sufficiently small. We fix an auxiliary parameter . Then, we apply Corollary 5.6 to find the constant which depends only on . Finally, we will assume, presently, that , and .
Let be the uniformly distributed probability measure on ; in particular is a -measure (with absolute constant), and . Apply Proposition A.1 to find a set of cardinality such that , where is the uniformly distributed probability measure on . Write , so is a -probability measure by Lemma 5.9. Since and , it follows form Corollary 5.6 that there exists a set of measure such that
| (5.13) |
Finally, write for all , and note that for all (this is a consequence of the general inequality ). Consequently, also , using . Therefore,
using Cauchy-Schwarz, and it follows from (5.13) that for at least one vector . On the other hand, note that is a union of left translates of , and recall from (2.1) that
Therefore, we have the upper bound
Since by assumption, the previous lower and upper bounds for are not compatible for small enough. A contradiction has been reached. ∎
6. Kakeya estimate of Guth, Wang, and Zhang
The purpose of this section is to prove Theorem 5.2. This will be based on the duality between horizontal lines and light rays developed in Section 4, and an application of a (reverse) square function inequality for the cone, due to Guth, Wang, and Zhang [10]. To be precise, we will not need the full power of this "oscillatory" statement, but rather only a Kakeya inequality for plates in [10, Lemma 1.4]. To introduce the statement, we need to recap some of the terminology and notation in [10]. This discussion follows [10, Section 1], but we prefer a different scaling: more precisely, in our discussion the geometric objects (plates and rectangles) of [10] are dilated by "" on the frequency side and (consequently) by on the spatial side.
Fix , and let
| (6.1) |
Let be the -neighbourhood of , and let be a finitely overlapping cover of by rectangles of dimensions , whose longest side is parallel to a light ray. The statements in [10] are not affected by the particular construction of , but in our application, the relevant rectangles are translates of dual rectangles of the -plates in Definition 4.16, with . Indeed, -plates are rectangles of dimensions tangent to , so their dual rectangles are plates of dimensions , also tangent to (this is because has opening angle , see Figure 3). For concreteness, we will use translated duals of -plates (as in Definition 4.16) to form the collection .
For each , let be a function with , and consider the square function
Then, [10, Lemma 1.4] contains an inequality of the following form:
| (6.2) |
To understand the meaning of the "partial" square functions we need to introduce more terminology from [10]. Fix a dyadic number (an "angular" parameter), and write . The -neighbourhood of the truncated cone can be covered by a finitely overlapping family of rectangles of dimensions
(Here agrees with , as defined above.) Consequently, the -neighbourhood of is covered by the rescaled rectangles
of dimensions . Note that the family coincides with (at least if it is defined appropriately), whereas consists of balls of radius . For every , the rectangles in are at least as large as those in , so we may assume that every is contained in at least one rectangle .
For and , let and be the dual rectangles of and (here the word "dual" refers to the common notion in Euclidean Fourier analysis, and not the duality in the sense of Proposition 4.22). Then both and are rectangles centred at the origin, with dimensions
respectively. The longest sides of both and remain parallel to a light ray on : this is again the convenient property of the "standard" cone with opening angle , see Figure 3. Of course, is an -plate in the sense of Definition 4.16, since the elements were defined as (translates of) duals or -plates.
The set turns out to be (essentially) a dilate of an -plate. For every , consider , which is a rectangle of dimensions
In particular, is an -plate, and hence larger than (or at least as large as) : if , then every translate of is contained in some translate of . We let be a tiling of by rectangles parallel to . Now we may finally define the "partial" square function :
| (6.3) |
We have now explained the meaning of (6.2), except the sum over "". In our notation, this means the same as summing over .
We are then prepared to prove Theorem 5.2.
Proof of Theorem 5.2.
Let , and let be a -set of -balls with -separated centres. In the statement of Theorem 5.2, it was assumed that , but for slight technical convenience we strengthen this (with no loss of generality) to for a small absolute constant . As in the statement of Theorem 5.2, let be the measure on with density
Following the discussion Section 4.1, and in particular recalling equation (4.15), Theorem 5.2 will be proven if we manage to establish that
| (6.4) |
assuming that are small enough, depending on . Recall that . To estimate the quantity in (6.4), notice first that
| (6.5) |
because for all . Write . Then, as we just saw,
The second inequality is based on (a) the definition of the measure , and (b) the observation that if and , then , and this forces (if was taken small enough). Finally, by Lemma 4.11, we have
Indeed, whenever for some , there exists a point , and then Lemma 4.11 implies that . Therefore, combining (6.4)-(6.5), it will suffice to show that for fixed, the inequality
| (6.6) |
holds assuming that we have picked (in the -set hypothesis for ) sufficiently small, depending on . We formulate a slightly more general version of this inequality in Proposition 6.7 below, and then explain in the remark afterwards why (6.6) is a consequence. This completes the proof of Theorem 5.2. ∎
Proposition 6.7.
For every , there exists such that the following holds for all . Let be a family of -balls contained in with -separated centres, and satisfying the following non-concentration condition for some :
| (6.8) |
Then,
| (6.9) |
Remark 6.10.
Why is (6.6) a consequence of (6.9)? In (6.6), we assumed that is a -set. This implies
Therefore, (6.8) is satisfied with constant . Hence (6.9) implies (6.6) if we choose and then sufficiently small.
We chose to formulate Proposition 6.7 separately because the "meaning" of (6.9) is easier to appreciate than that of (6.6): namely, if all the sets had a disjoint intersection inside , then the left hand side of (6.9) would be roughly . Thus, (6.9) tells us that under the non-concentration condition (6.8), the sets are nearly disjoint inside , at least at the level of -norms.
Proof of Proposition 6.7.
By the discussion in Section 4.2, the intersections are essentially -plates – rectangles of dimensions tangent to . More precisely, for every , let be a -plate (as in Definition 4.16) with the property
This is possible by first applying Proposition 4.22 (which yields a modified -plate containing ), and then the first inclusion in (4.21), which shows that the intersection of the modified -plate with is contained in a -plate . Now, we will prove (6.9) by establishing that
| (6.11) |
Every plate has a direction, denoted : this is the direction of the longest axis of , or more formally the real number "" associated to the line "" in Definitions 4.16. By enlarging the plates slightly (if necessary), we may assume that their directions lie in the set : this is because if two plates coincide in all other parameters, and differ in direction by , both are contained in constant enlargements of the other (this is not hard to check). The reason why we may restrict attention to is that all the plates were associated to the balls , and in fact the -coordinate of the centre of determines the direction of (see (4.10)).
We next sort the family according to their directions:
where . Thus, for fixed, the plates in are all translates of each other. Also, the plates in for a fixed have bounded overlap: this follows from the assumption that the balls in have -separated centres, and uses Lemma 4.33 (the plates with a fixed direction correspond precisely to Heisenberg balls whose -coordinates are, all, within "" of each other).
Write , thus , and recall the truncated cone from (6.1). Since the plates are translates of each other, they all have a common dual rectangle of dimensions . The rectangle is centred at , but we may translate it by in the direction of its longest -side (a light ray depending on ) so that the translate lies in the -neighbourhood of . Committing a serious abuse of notation, we will denote this translated dual rectangle again by "", and the collection of all these sets is denoted . This notation coincides with the discussion below (6.1). There is a -to- correspondence between the directions and the rectangles defined just above, so the notational inconsistency should not cause confusion.
We gradually move towards applying the inequality (6.2) of Guth, Wang, and Zhang. The next task is to define the functions and . Fix , , and let be a non-negative Schwartz function with the properties
-
(1)
,
-
(2)
has rapid decay outside ,
-
(3)
.
Here "rapid decay outside has" the usual meaning: if denotes a -times dilated, concentric, version of , then for all (and for any ). Then, define the function
Here is a modulation, depending only on , such that
Now the function satisfies all the assumptions of the inequality (6.2), so
| (6.12) |
Recall the notation on the right hand side, in particular that only runs over dyadic rationals, and the definition of the "partial" square function from (6.3). The rectangles are -plates with . In particular, every is essentially the -dual of a Heisenberg -ball: this will allow us to control by applying the non-concentration condition (6.8) between scales and .
By definition,
| (6.13) |
Above, and in the sequel, the notation means that for every , there exists a constant such that . In (6.13), the final "" inequality follows easily from the rapid decay of the functions , and the bounded overlap of the plates for fixed.
For , each plate is contained in some translate of (this was discussed above (6.3)), but this translate may not be . Let be an -plate which is concentric with . We then decompose the right hand side of (6.13) as
| (6.14) |
Since each is contained in element of the tiling (consisting of translates of ) every plate with is far away from : more precisely, . By the rapid decay of outside , this implies that on , and therefore the second term of (6.14) is bounded by, say, .
We then focus on the first term of (6.14), and we first note that
| (6.15) |
since . So, we need to find out how many -plates are contained in . Since is an -plate, it follows from the second inclusion (4.21), combined with the second inclusion in Proposition 4.22, that
for some , and for some absolute constant . On the other hand, the plates , , were initially chosen in such a way that . Thus, whenever , we have
This implies by Proposition 4.30 that , where possibly was inflated by another constant factor. Thus,
Using (6.8), this will easily yield useful upper bounds for .
To make this precise, we sort the sets "" appearing in (6.12) according to the "richness"
| (6.16) |
For fixed, we choose a (dyadic) value such that
| (6.17) |
Here "" hides a constant of the form . Let be the collection of sets "" appearing on the right hand side, and let be the subset of the original -balls which are contained in some ball , . Then, evidently,
| (6.18) |
The factor "" arises from the fact that while distinct sets "" are the duals of essentially disjoint Heisenberg -balls, the inflated balls only have bounded overlap, depending on the inflation factor .
Now, for , we may estimate (6.15) as follows:
(In this estimate, we have omitted the term "" from the second part of (6.14), because this term will soon turn out to be much smaller than the best bounds for what remains.) Plugging this estimate into (6.17), and observing that , we obtain
Notably, this estimate is independent of "" and the parameter "", so we may finally deduce from (6.12) that
Since and was arbitrary, this implies (6.9) by renaming variables, and the proof of Proposition 6.7 is complete. ∎
7. Proof of Theorem 1.7
We recall the statement:
Theorem 7.1.
Let be a Borel set with . Then, for almost every . Consequently, for almost every .
Proof.
The lower bound for follows immediately from the lower bound for , combined with a general inequality between Hausdorff dimensions relative to Euclidean and Heisenberg metrics of subsets of , see [1, Theorem 2.8]. So, we focus on proving that for almost every .
The first steps of the proof are standard; similar arguments have appeared, for example the deduction of [16, Theorem 2] from [16, Theorem 1]. So we only sketch the first part of the proof, and provide full details where they are non-standard. First, we may assume that , and we may assume, applying Frostman’s lemma, that for some Borel probability measure satisfying for all and .
We make the counter assumption that there exists such that
By several applications of the pigeonhole principle, this assumption can be applied to find the following objects for any , and for arbitrarily small :
-
(1)
A Borel subset of length .
-
(2)
For every a collection of Euclidean -discs , contained in .
-
(3)
If and , then
(7.2)
We claim that (1)-(3) violate Theorem 5.11 if are small enough. To this end, we first need to construct a relevant -set of (Heisenberg) -balls contained in . Morally, this collection is a -approximation of . More precisely, we need to decompose to the following subsets:
where runs over dyadic rationals with . By one final application of the pigeonhole principle, and recalling (7.2), one can find a fixed index such that
| (7.3) |
for all , where . In particular, . Then, we let be a (Vitali) cover of by finitely overlapping Heisenberg -balls with -separated centres. Note that . Using the definition of , and the Frostman condition for , it is now easy to check that is a -set of -balls, where is roughly the Frostman constant of .
Finally, from (7.3) and , we deduce that if , then intersects elements of , since
Write , thus . We now arrive at the point where it is crucial that the elements of are Euclidean -discs. Namely, if , then for some . Then, because is a Euclidean -disc, and the Euclidean diameter of is , we may conclude that . This could seriously fail if were a disc in the metric . Now, however, we see that
and in particular for all . This violates the conclusion of Theorem 5.11, and the proof of Theorem 7.1 is complete. ∎
Appendix A Completing -sets to -sets
In this section, we use the following notation for the -truncated -dimensional Riesz energy of a Radon measure on :
where . We also recall that is the Heisenberg convolution of and , that is, the push-forward of under the group operation .
Proposition A.1.
Let with , and let . Let be a Borel probability measure on with . Then, there exists a set with such that the uniformly distributed (discrete) measure on satisfies
where for some absolute constant .
Proof.
Let be a grid of Euclidean -separated lattice points in . Then . Let be a random set, where each point of is included independently with probability . In particular, . While we use the symbol "" to index the elements in the underlying probability space, no explicit reference to this space will be needed. Let be the random measure
We claim that
| (A.2) |
for some . In this argument, the notation "" hides a constant of the form . The inequality (A.2) will complete the proof of the proposition, because with probability (for small enough), and therefore, by Chebychev’s inequality, for some "" with .
To prove (A.2), it clearly suffices to establish that
| (A.3) |
By definition of , we have
We consider the expectations of and separately. The former one is simple, using that :
recalling that . To handle the expectation of , we note that and are independent events for , hence
where "" runs over dyadic rationals. Since the product "" is not commutative, in general , so the set is not contained in a -ball of radius around . This is the key inefficiency in the argument, and causes the restriction : under this restriction, it actually suffices to note that is contained in a Euclidean -ball. To see this, note that if satisfies with , then
Here is contained in a Euclidean ball of radius (using ). The same remains true after the right translation by , because (by assumption), and the right translation is Euclidean Lipschitz with constant depending only on .
Now, since a Euclidean -ball contains points of , we see that
where in the final inequality we used again that . This completes the proof of (A.3), and therefore the proof of the proposition. ∎
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