Dimension Distortion by Right Coset Projections in the Heisenberg Group

We study the family of vertical projections whose fibers are right cosets of horizontal planes in the Heisenberg group, $\mathbb{H}^n$. We prove lower bounds for Hausdorff dimension distortion of sets under these mappings, with respect to the Euclidean metric and also the natural quotient metric which we show behaves like the Euclidean metric in this context. Our bounds are sharp in a large part of the dimension range, and we give conjectural sharp lower bounds for the remaining range. Our approach also lets us improve the known almost sure lower bound for the standard family of vertical projections in $\mathbb{H}^n$ for $n \geq 2$.


Introduction
The study of dimension distortion by projections dates back to J. Marstrand's 1954 paper [18]. Among many other things, it was shown that for an analytic set A Ă R 2 , dim P θ pAq " mintdim A, 1u for H 1 -almost all θ P r0, πq, where P θ : R 2 Ñ ℓ θ is the orthogonal projection onto the line with terminal angle θ. Moreover, it was shown that if dim A ą 1 then H 1 pP θ pAqq ą 0 for H 1 -almost all θ P r0, πq. Over time, this result has been expanded and generalized in many directions. For instance in [17], R. Kaufman introduced a potential theoretic approach that streamlined Marstrand's proof, and using this approach P. Mattila generalized the result to higher dimensions [19]. The general result, including the Besicovitch-Federer characterization of unrectifiability ( [7], [12]), is stated in the following theorem. Theorem 1.1. Let A Ă R n be an analytic set of dimension s.
(1) If s ď m, dim P V pAq " s for almost every m-dimensional subspace V .
(2) If s ą m, H m pP V pAqq ą 0 for almost every m-dimensional subspace V .
(3) If s ą 2m, IntpP V pAqq ‰ ∅ for almost every m-dimensional subspace V . Moreover, in the case where s " m and with the added hypothesis that H m pAq ă 8, A is purely m-unrectifiable if and only if H m pP V pAqq " 0 for almost every mdimensional subspace V .
Analogous, but in some cases weaker, results have been obtained when projections are restricted to a subfamily of planes [3,16,11,24,9,25,14]. In [26] the authors introduced the concept of transversal families of maps thus giving a vast generalization of Theorem 1.1 which extended the result to many more families of mappings. The problem has also been studied outside of the Euclidean setting, specifically in the Heisenberg group, in [2,3]. There, the story is far from over. Two distinct families of "projections" arise naturally in this context, known as homogeneous projections. Dimension distortion by one of these families, that of horizontal projections, can be tackled using transversality, but the other family, that of vertical projections, is not transversal in the sense of Peres and Schlag and is otherwise quite difficult to work with. Improving the known dimension distortion bounds in this context continues to be an active area of research with improvements being made recently in [15]. In this paper we continue the work in this direction by studying another natural, yet unstudied, family of projections in the Heisenberg group. Our approach also improves the known dimension distortion bound for the standard family of homogeneous projections studied in [3].
The nth Heisenberg group is defined as the manifold H n :" R 2nˆR with typical point denoted by pz, tq " px 1 , . . . , x n , y 1 . . . , y n , tq where for j " 1, . . . , n, z j " x jì y j . As such, we will identify C n with R 2n (and e.g. write iz for pointwise scalar multiplication of z P C n by i). We endow this manifold with the group law pz, tqp w, sq " pz`w, t`s`1 2 ωpz, wqq, where w " pu 1 , . . . , u n , v 1 , . . . , v n q, and ωpz, wq " ř n j"1 px j v j´yj u j q. This group law makes H n a Lie group with left invariant vector fields For any given j, rX j , Y j s " T , so H " spantX j , Y j : j " 1, . . . , nu forms a bracket generating distribution. We say an absolutely continuous curve γ : r0, 1s Ñ H n is horizontal if 9 γpsq P H γpsq for a.e. s P r0, 1s.
By declaring tX j , Y j : j " 1, . . . , nu to be orthonormal, we can compute the (horizontal) length of γ in the usual way. We will denote the length of γ by |γ|. The bracket generating condition enables the definition of a Carnot-Carathéodory distance in all of H n via d cc pp, qq " inft|γ| : γ is horizontal, and γp0q " p, γp1q " qu.
The Korányi gauge pz, tq 4 H n " |z| 4`1 6t 2 also gives a left invariant metric (known as the Korányi metric) given by d H n pp, qq " q´1˚p . These two metrics are bi-Lipschitz equivalent.
For r ą 0 the non-isotropic dilations δ r pz, tq " prz, r 2 tq give H n a homogeneous structure. This enables the definition of homogeneous subgroups as subgroups which are closed under dilations. These subgroups come in two kinds, those contained in C nˆt 0u (horizontal), and those containing the entire t-axis (vertical). The t-axis is a homogeneous subgroup, one without a complementary horizontal subgroup. The horizontal subgroups Vˆt0u coincide with isotropic subspaces V of C n , and their (Euclidean) orthogonal complements V KˆR are vertical subgroups (here an isotropic subspace means one on which the symplectic form ω vanishes identically). We denote the Grassmannian of isotropic m-planes in R 2n as G h pn, mq, and for V P G h pn, mq, we denote the corresponding horizontal and vertical subgroups by V and V K respectively. For each V P G h pn, mq, V K is a normal subgroup of H n , and we have a semi-direct splitting H n " V˙V K . Since the group V K is normal, the splitting can also be taken to be H n " V K¸V . These splittings induce projection maps P V onto the horizontal subgroup V, and P R V K , P L V K onto the vertical subgroup V K . Here P R V K is induced by the first mentioned splitting, and its fibers are right cosets of the subgroup V. In the same way, P L V K is induced by the second splitting and its fibers are left cosets of the horizontal subgroup V. Turns out, P V agrees with the Euclidean orthogonal projection onto the subspace V , while P R V K , and P L V K can be defined via the group law by P R V K ppq " P V ppq´1p, P L V K ppq " pP V pP q´1. Since the group law is non-commutative, these two maps are inherently different, although they are related by the equation It is important to note that given a set A Ă H n dim H n A ‰ dim H n p´Aq in general. It is therefore expected that these maps behave differently when it comes to dimension distortion.
The group Upnq of complex unitary matrices, which may be identified as a subgroup of Op2nq, preserves the symplectic form ω (see [20,Chapter 3]). This group acts smoothly and transitively on G h pn, mq, and each R P Upnq induces an isometry of H n given by Rpz, tq " pRz, tq. Therefore, for any two horizontal subgroups V and V 1 there is an R 0 P Upnq such that V " R 0 V 1 . Since Upnq has a unique probability Haar measure, the space G h pn, mq inherits a unique Upnq-invariant probability measure, which we denote by µ n,m . This in turn allows us to put a measure on the set of horizontal (resp. vertical) subgroups of Hausdorff dimension m (resp. 2n`2´m) in H n , specifically, one simply uses the measure µ n,m by appealing to the aforementioned correspondence between horizontal (resp. vertical) subgroups and G h pn, mq.
The vertical projections P L V K , together with horizontal projections, have been heavily studied in the context of Hausdorff dimension distortion ( [2], [3], [10], [15]). These projections also play a pivotal role in the theory of rectifiable sets in H n ( [23]). Here we intend to initiate the study of the projection P R V K in the context of dimension distortion. Whereas the fibers of the map P L V K are horizontal lines, the fibers of P R V K are not horizontal. It is therefore not very natural to consider P R V K as a map from pH n , d cc q to pV K , d cc t V K q. In H 1 , the maps P R V K have already been studied in other contexts (see for instance [1]) where a natural metric arises on the image of P R V K . We study dimension distortion in the context of this, "more natural", metric by first generalizing it to higher dimensions. Our main result is as follows.
Theorem 1.2. For 1 ď m ď n and any Borel set A Ď H n , for µ n,m -a.e. V P G h pn, mq, and Here, d VzH n refers to this aforementioned "more natural" metric on V K while dim E and dim H n refer to the Hausdorff dimension with respect to the Euclidean and Heisenberg metrics, respectively (see [6]). Our main idea is to obtain a projection theorem in the Heisenberg group by first considering the Euclidean metric on both sides and then applying some kind of "dimension comparison principle". This is natural for right coset projections because the resulting bound obtained is sometimes sharp. We remark that the Euclidean-Euclidean dimension distortion problem for vertical projections in H seems to have been first posed in [21, p. 296]. At least one instance of applying Euclidean methods and dimension comparison to projection bounds in the Heisenberg group can be found in the proof of Proposition 4.9 in [3].
By dimension comparison, Theorem 1.2 leads to the following almost sure dimension bound for the standard (left-coset) projection problem.
for µ n,m -a.e. V P G h pn, mq.
Previously, the best known almost sure lower bound for this problem (in H n with n ą 1 and dim H n A ď m`2) was dim H n P L V K pAq ě mintdim H n A, 1u for µ n,m -almost all V P G h pn, mq. This bound also holds when n " 1, though there it is not the best known. The best known universal lower bound was From this, the best possible almost sure lower bound was for µ n,m -almost every V P G h pn, mq. Therefore, Theorem 1.3 improves this almost sure lower bound in the range dim H n A P r2, 2n`1s. The new lower bound reads if dim H n A P r2n´m`1, 2n`1s 2pdim H n A´n´1q´m if dim H n A P r2n`1, 2n`2s, for µ n,m -almost every V P G h pn, mq.
For n ą 1, we do not know if the lower bounds in Theorem 1.2 are sharp for dim E A ě 2n´m and dim H n A ě 2n`1´m, but we suspect the answer is no. For dim H n A ą 2 we predict the lower bound dim H n A´1 to hold up to dim H n A " 2n`2´m; the example in the proof of Theorem 1.2 shows this would be sharp. The conjectured lower bounds are given below; in all cases it is assumed that 1 ď m ď n. Conjecture 1.4 (see [2,Conjecture 1.5] for the case n " 1). Let A Ď H n be a Borel set. If dim H n A ď 2n`2´m then Finally, in the first Heisenberg group H there is a small improvement possible to Theorem 1.2, which we show in Section 4. With Euclidean metrics on each side, Corollary 4.3 is a better a.e. lower bound than Theorem 1.2 for dim E A P p1, 5{2q.

Right coset projections in H n
In this section we will first introduce the Grushin plane, which will come back later in connection with right coset quotient spaces. Then we will describe the right coset quotient space by vertical subgroups together with the corresponding vertical projections. Finally we will restrict to the case of vertical subgroups of co-dimension one where we have a clear description of the metric structure of the space and the aforementioned connection with the Grushin plane arises. It is worth mentioning that the connection between the Heisenberg group and the Grushin plane has been studied before (see for instance [1], [27] and [13, (3) p.293]).
In this section we only consider the projections P R V K which we will simply denote by P V K . In addition, for 1 ď m ď n, the notation H n´m will be frequently used. It is therefore important to emphasize that this notation signifies the pn´mqth Heisenberg group, C n´mˆR , with all of its structure. In particular, when n " m, H n´m is simply the "t-axis", C 0ˆR " R, with standard addition and metric d H n´m " 2d 1{2 E .
2.1. The Grushin Plane. The Grushin plane is the manifold G " R 2 with vector fields where pv, τ q P R 2 . These vector fields span the whole tangent space at every point outside of the singular set tv " 0u, and by taking them to be orthonormal there, we get a line form Bτ , which allows us to extend this metric to a Carnot-Carathéodory path distance in all of R 2 . The resulting metric, denoted by d G , turns G into a non-equiregular sub-Riemannian manifold whose horizontal curves are curves that have horizontal tangent at every point of intersection with the critical line. That is to say, γ : r0, 1s Ñ G is horizontal if there exist integrable functions a and b such that 9 γpsq " apsqT`bpsqV, for a.e. s P r0, 1s. The length of γ is then given by If we write γpsq " pvpsq, τ psqq, a more explicit formula for the length is For each t 0 the vertical translation map pv, tq Ñ pv, t`t 0 q is an isometry of G. This can also be seen as a non-transitive group action by R whose orbits are vertical lines, in particular, the orbit of 0 is the critical line v " 0. One interesting property of the Grushin metric, that will come back later in the discussion, is that the restriction of the distance to the critical line is comparable to the square root of the Euclidean distance. Therefore, this "copy" of R is embedded into G in a "snowflaked" way. In contrast, the restriction of the distance to any other vertical line is Riemannian.
2.2. The right coset quotient space. For 1 ď m ď n, given V P G h pn, mq we consider the quotient space of right cosets of V in H n , VzH n :" tVp : p P H n u , endowed with the quotient distance d VzH n pVp, Vp 1 q " inf td cc pqp, p 1 q : q P Vu .
There is a unique way to write elements of VzH n as Vq with q P V K . Therefore VzH n is identified with V K by the map Vq Þ Ñ q. This map coincides with the map on VzH n induced by P V K , that is P V K pVpq " tP V K ppqu.
Lemma 2.1. For each fixed V, the map An upper bound is found by choosing a specific q P V. In particular, choosing q " P V pp 1 q´1P V ppq, and appealing to the left invariance of d cc we see that, Denoting by π W the Euclidean orthogonal projection onto W , an explicit formula for the projection is given by he space VzH n inherits a rich structure from H n which allow us to have a more intuitive understanding of the space.
The unitary group, Upnq, acts smoothly and transitively on G h pn, mq and isometrically on H n via pz, tq Ñ pRz, tq, pR P Upnqq, therefore understanding the metric properties of V 0 zH n for a fixed V 0 will get us the same properties for VzH n in general. Hence, to simplify computations, fix the horizontal subgroup V " V 0 :" tpx 1 , . . . , x m , 0, . . . , 0q : x j P Ru, for the rest of this section. This gives us . , x n , y 1 , . . . , y n , tq : x j , y j , t P Ru. With this concrete setting, we discuss some of the symmetries of the space VzH n .
Homogeneous dilations. The space VzH n admits homogeneous dilations. Although these dilations are defined on VzH n we abuse notation using the same symbol as for the Heisenberg dilations since the dilations on VzH n are nothing more than the dilations on H n that factor through the quotient map. For each r ą 0 the map δ r : VzH n Ñ VzH n given by δ r p0, . . . , 0, x m`1 , . . . , y n , tq "`0, . . . , 0, rx m`1 , . . . , ry n , r 2 t˘, is homogeneous of degree 1 with respect to d VzH n . Indeed: d VzH n pδ r ppq, δ r pp 1 qq " inf qPV d cc pqδ r ppq, δ r pp 1 qq " r inf qPV d cc pδ 1{r pqqp, p 1 q " rd VzH n pp, p 1 q.
The last equality follows from the fact that V is homogeneous (so that δ 1{r pqq P V).
Group action by H n´m . We embed H n´m in H n by the map ξ Þ Ñ p ξ given by, where in the right hand side the first m coordinates and coordinates n`1 through n`m are all zero. With this notation we can see that H n´m acts on H n by "left translation" via the map L ξ p " p ξp.
To see that this action is isometric, note that for each ξ P H n´m , p ξ commutes with elements of V. Indeed, writing q " pz, 0q P V and p ξ " p p w, τ q, it is not hard to see that ω p p w, zq " 0. Because of this, This action is smooth with respect to the quotient topology but it is not transitive. For a point p0, . . . , 0, x m`1 , . . . , x n , y 1 , . . . , y n , tq P V K its orbit consists exactly of all other points of the form p0, . . . , 0, x 1 m`1 , . . . , x 1 n , y 1 , . . . , y m , y 1 m`1 , y 1 n , t 1 q. Therefore, the orbit space is parametrized by R m .
Group action by Upn´mq. Similarly, we embed Upn´mq into Upnq via the map R Þ Ñ r R given for each z " px 1 , . . . , x n , y 1 , . . . , y n q P R 2n by r Rz " r z.
In this way Upn´mq acts on VzH n via p Þ Ñ p Rp :"´r Rz, t¯where p " pz, tq P V K » VzH n . Once again, it is not hard to check that this action, as an action naturally extended to all of H n , fixes V pointwise. Therefore p Rpqpq " q p Rp for each q P V and p P V K . Since Upnq acts isometrically on H n , it follows that Like the H n´m action, the action by Upn´mq is smooth but not transitive. The orbit of a point p0, . . . , 0, x m`1 , . . . , x n , y 1 , . . . , y n , tq P V K consists of all other points of the form p0, . . . , 0, x 1 m`1 , . . . , x 1 n , y 1 , . . . y m , y 1 m`1 , y 1 n , tq. Therefore, the orbit space is parametrized by R m`1 .
The group action by H n´m reveals that there are "R m many" copies of the set H n´m embedded in V K in a natural way. More precisely, using the notation p " px 1 , x 2 , y 1 , y 2 , tq P R mˆRn´mˆRmˆRn´mˆR " H n , for a fixed r y P R m we denote by U r y the orbit U r y " tL ξ p0, 0, r y, 0, 0q P H n : ξ P H n´m u. The map H n´m Ñ U r y given by px, y, tq Ñ p0, x, r y, y, tq gives a natural embedding of the set H n´m into V K .
Proposition 2.2. The restrictions of d VzH n and d cc to U r 0 , are bi-Lipschitz equivalent.
Proof. For any x 1 P R m , x 2 , y 2 P R n´m , and t P R one can check directly from the formula for the Koranyi norm that, Now, as mentioned earlier, it is easy to check that ωpV, U r 0 q " 0 so that V and U r 0 commute, and moreover, for q P V and p P U r 0 , qp " q`p. In particular, if p, p 1 P U r 0 it follows that where the first equality in the last line follows from (2.8). This completes the proof of the proposition.
Corollary 2.3. The map ι : pH n´m , d cc,H n´m q Ñ pV K , d VzH n q given by ιpx, y, tq " p0, x, 0, y, tq is a bi-Lipchitz embedding.
Proof. It is clear that ι : Proposition 2.2 and its corollary, do not hold for r y ‰ 0. In particular, for r y ‰ 0, the natural bijection of H n´m onto the orbit U r y is not a bi-Lipschitz, embedding. Indeed, if r y ‰ 0 and p " p0, x, r y, y, 0q, q " p0, u, r y, v, 0q P U r y , we have whereas, In particular, choosing r x "´1 2 px¨v´y¨uq r y |r y| 2 gives the upper bound Comparing with (2.9) one sees that d V 0 zH n t U r y cannot be bi-Lipschitz equivalent to d H n t U r y , and therefore to d cc t U r y . We expect the space VzH n to behave in an analogous way to the Grushin plane, G, in that the metric should be Riemannian away from the critical subspace U r 0 and extend as a Carnot-Carathéodory metric to U r 0 . We were unable to prove this in general, so it remains an interesting problem to check if pV K , d VzH n q is isometrically equivalent (or at least bi-Lipschitz equivalent) to a non equi-regular Carnot-Carathéodory space. In the specific case m " 1 this is exactly true as we will see in the following section were we state this formally and give a sketch of the proof.

2.3.
Vertical subgroups of co-dimension one. Consider the manifold R 2n , with typical point denoted pw, u 1 , . . . , u n´1 , w 1 , . . . , w n´1 , τ q, and frame comprised of the vector fields Bτ . These vector fields span the entire tangent plane at every point outside of the critical p2n´1q-plane tw " 0u, thus by declaring it orthonormal there, it induces a Riemannian distance on R 2n ztw " 0u. Moreover, since rU j , W j s " rT, W s " B Bτ , this metric can be extended to a Carnot-Caratheodory metric, d ∆ , on all of R 2n . Note that in the case n " 1, the frame ∆ consist only of the vector fields V and T and therefore the space pR 2 , d ∆ q coincides with the Grushin plane.
Sketch of proof. It is clear that, as sets, V K and R 2n can be identified, so we consider the map P V K as a map from H n to R 2n . Firstly, we use the analytic change of variables in H n Ψpz, tq " rz, ts " pz, t`1 2 Under this change of variables the horizontal vector fields become Bt , j " 2, . . . , n, and the projection map becomes The differential of this map is easily computed to be the constant matrix where I is the p2nqˆp2nq identity. Hence, the push forward of the horizontal vector fields are Note that these coincide exactly with (2.10), therefore if Γ : It follows that γ is horizontal in pR 2n , d ∆ q and moreover, This tells us that given p, p 1 P V K " R 2n , every H n -horizontal path between Vp and Vp 1 induces a ∆-horizontal path between p, p 1 P R 2n of the same length. Thus Now we aim to show that every horizontal path in pR 2n , ∆q between p, p 1 has a H n -horizontal lift between Vp and Vp 1 of the same length. This would imply d VzH n pp, p 1 q ď d ∆ pp, p 1 q and complete the proof.
It follows that Φ V K pΓq " γ and so Γ is a horizontal path in H n between the fibers Φ´1 V K p0, γp0qq and Φ´1 V K p0, γp1qq. Furthermore, and this completes the proof.
Note that whenever n ą 1 the vector fields tU j , W j : j " 1, . . . , n´1u give rise to the embedded copy of H n´1 in pV K , d VzH n q that was mentioned in last section. On the other hand, as mentioned above, when n=1 the frame ∆ only consist of V and T , and the Carnot-Caratheodory manifold pV K , d VzH n q is exactly the Grushin plane G with the embedded copy of "H 0 " corresponding to the critical line. This last fact has been well known and used in conjunction with the right coset projections in the first Heisenberg group to solve certain iso-perimetric problems in the Grushin plane by projecting Heisenberg geodesics via P V K ([1]).

Dimension distortion by right coset projections in H n
We now have the appropriate setup to study dimension distortion by right coset projections. We have a family of 1-Lipschitz maps P V K : pH n , d cc q Ñ pV K , d VzH n q : V P G h pn, mq ( and would like to study the generic dimension of the sets P V K pAq for a given Borel set A Ă H n . First we note that since the maps are Lipschitz, the upper bound dim VzH n P V K pAq ď dim H n pAq holds trivially for all V. Therefore, our main result focuses on almost sure dimension lower bounds. As we will see in the proof of the main result, lower bounds for the Euclidean Hausdorff dimension of projections will help us obtain lower bounds for their dimension with respect to the metric d VzH n .
For any Borel subset A of a complete separable metric space pX, dq, the Hausdorff dimension dim A of A can be characterised using energy: dim A is the supremum over all s ě 0 such that there exists a compactly supported probability measure µ on A with I s pµ, dq :" ż ż dpx, yq´s dµpxq dµpyq ă 8.
The first four lemmas will show that for dimension lower bounds the a.e. behaviour of projections with respect to the right coset metric is the same as with respect to the Euclidean metric. Proof. Fix R ą 0 and pz, tq, pζ, τ q P V K X B E p0, Rq. To prove d E ppz, tq, pζ, τ qq À R d VzH n ppz, tq, pζ, τ qq, it suffices to show that (3.13) |z´ζ|`|t´τ | À R |z`w´ζ|` t´τ`1 2 ωpz, ζq´1 2 ωpz`ζ, wq

1{2
, uniformly for all w P V . If |t´τ | ď 2R|z´ζ| then (3.13) follows from orthogonality, using only the first term in the right hand side. Hence it may be assumed that If |w| ě |t´τ | 4R then (3.13) again follows from orthogonality, so it may be assumed that Thus Taking square roots gives (3.13), and therefore proves the lemma.
The following lemma gives a sufficient condition under which the preceding inequality can be reversed.
(i) For any Borel set A Ď H n with dim H n A ą α, The implication (ii) ñ (i) follows directly from Lemma 3.1, so assume that (i) holds. Let A Ď H n be a compact set with dim H n A ą α ě 2. Let µ be a Borel probability measure on A with µpB H n ppz, tq, rqq À r s for all pz, tq P H n and r ą 0, By a similar covering argument to the proof of Theorem 1.1 in [3], where the inner integral is bounded using the inequality from the proof of Theorem 1.2 in [3]. Let U be a nonempty open subset of G h pn, mq such that there exists a continuously varying orthonormal basis tv 1 pV q, . . . , v m pV qu for V as V varies over U (which exists e.g. by Gram-Schmidt). By covering G h pn, mq with a finite number of such sets, it will suffice to show that dim E P V K pAq ě β for a.e. V P U.
Hence for each k there exists V k P U pkq 0 and a Borel set B k Ď A with µpB k q Á 1 and |π iV k pzq| Á δ for all pz, tq P B k .
Therefore for each k there exists j " jpkq P t1, . . . , mu, σ " σpkq P t0, 1u and a Borel set A k Ď A such that (3.22) µpA k q Á 1 and ω pz, p´1q σ v j pV k qq Á δ for all pz, tq P A k .
Since each V is isotropic, it then follows from Lemmas 3.1 and 3.2 that for µ n,m -a.e. V P U 0 . But µ n,m pU 0 q ě µ n,m pUq´?ǫ by (3.19), so letting ǫ Ñ 0 and covering G h pn, mq with a finite number of such sets U gives for a.e. V P G h pn, mq. This proves that (i) and (ii) are equivalent.
The preceding lemma and the following one actually hold for α ě 0, but for small α this follows from Theorem 1.2 (since the a.e. lower bounds for dimension that follow from Theorem 1.2 are the same in the smaller range of α, and by Theorem 1.2 they are both sharp). The proof of the following lemma is omitted since it is virtually identical to the previous one, except that s´1 is used in (3.16) instead of s´2.
(1) For any Borel set A Ď H n with dim E A ą α, dim VzH n P R V K pAq ě β for a.e. V P G h pn, mq.
(2) For any Borel set A Ď H n with dim E A ą α, dim E P R V K pAq ě β for a.e. V P G h pn, mq. We restate Theorem 1.2 here. Theorem 1.2. For 1 ď m ď n and any Borel set A Ď H n , , dim E A´mu for µ n,m -a.e. V P G h pn, mq, and Proof of Theorem 1.2. The cases P L V K and P R V K in (3.23) are equivalent since P L V K ppq " P R V K p´pq. For the remainder of the proof, the notation P V K will therefore denote P R V K . The quotient distance on V K is defined through the identification of V K with VzH n explained in Section 2.2; the formula is given by Since the metric d H n is bi-Lipschitz equivalent to d cc , one can set and trivially obtain that d VzH n and d 1 VzH n are bi-Lipschitz equivalent. For ease of computation we use d 1 VzH n instead of d VzH n , and to simplify notation we denote d 1 VzH n by d VzH n as well. It may be assumed without loss of generality that A is bounded. Let µ be a measure on A with Euclidean s-energy I s pµ, d E q ă 8, where s :" min tdim E A, 2n´mu´ǫ for an arbitrarily small ǫ ą 0. Assume s ą 0 without loss of generality. By Fubini, the average energy of the pushforward measure is ż To prove the Euclidean lower bound in the first part of the minimum of (3.23), it suffices to show (3.25) ż G h pn,mq d E pP V K pz, tq, P V K pζ, τ qq´s dµ n,m pV q À d E ppz, tq, pζ, τ qq´s.
The first half of this proof will be essentially the same as the proof of Theorem 1.2 in [3]. Let Bp0, Rq be a Euclidean ball containing A. If |z´ζ| ě |t´τ | 4R , then ż G h pn,mq d E pP V K pz, tq, P V K pζ, τ qq´s dµ n,m pV q À ż G h pn,mq |π V K pzq´π V K pζq|´s dµ n,m pV q À s |z´ζ|´s (since s ă 2n´m) (3.26) À R d E ppz, tq, pζ, τ qq´s; the Euclidean inequality used in (3.26) is explained in [3, pp. 584-585], and has a fairly straightforward proof. This proves (3.25) in the case where |z´ζ| ě |t´τ | 4R . In the second case with |z´ζ| ă |t´τ | 4R , Cauchy-Schwarz gives This proves the Euclidean lower bound for the first term in the maximum of (3.23), which finishes the proof of (3.23) in the case dim E A ď 2n. The lower bound dim E P V K pAq ě dim E pAq´m actually holds for all V K , provided dim E A ą m`1. Since the previous bound is stronger whenever dim E A ď 2n, we may assume, without loss of generality, that dim E A ą 2n. In particular dim E A ą m`1. From here the proof follows the same lines as the proof of the same lower bound for the H n -dimension of (left coset) vertical projections from [3, Theorem 1.4]. Given V P G h pn, mq the set tU P G h pn, mq : is open, nonempty and in particular has positive µ n,m measure. This, together with Theorem A.1, lets us pick for ǫ ą 0, U P G h pn, mq and u P U such that the map π V K t U K : U K Ñ V K is injective, and dim E rA X pU K˚u qs ě dim E A´m´ǫ.
For this particular choice of U and u, we will see that P V K t U K˚u: U K˚u Ñ V K is a locally bi-Lipschitz bijection with respect to the Euclidean norm. First we show injectivity. For any q P pU K˚u q, there exists a unique w U K P U K and s ą 0 such that q " pw U K , sq˚pu, 0q. Let q " pw U K , sq˚pu, 0q P pU K˚u q and q 1 " pz U K , tq˚pu, 0q P pU K˚u q be such that P V K pqq " P V K pq 1 q. Then we have The first coordinate tells us that π V K pz U K`uq " π V K pw U K`uq which says π V K pz U K q " π V K pw U K q. By our choice of U P G h pn, mq we get that z U K " w U K . Similarly, the second coordinate gives us that t " s so injectivity follows. To see that the map is surjective, for pz, tq P V K put ζ " pπ V K t U K q´1pz´π V K puqq`u, and τ " t`1 2 ωpπ V pζq, π V K pζqq. It follows that pζ, τ q P U K˚u and P V K pζ, τ q " pz, tq. This shows that the map is surjective, but also gives us a formula for the inverse which shows this inverse map is smooth. Hence P V t U K˚u is a smooth map with a smooth inverse, and it is therefore locally bi-Lipschitz with respect to the Euclidean metric. By the choice of U, Since ǫ can be chosen arbitrarily small, this proves the lower bound in (3.23 This comparison principle, as stated here, appears in [3, Eq. 1.4], see [4] for the original proof in H and see [5] for the proof in the more general case of Carnot groups. The sharpness of the Euclidean lower bound in (3.23) will be deduced from the sharpness of the Heisenberg lower bound in (3.24). The sharpness of (3.24) for dim H n A ď 2n`1´m will be proved in two separate cases. For an example with any Heisenberg dimension in the range r0, 2s, let α P r0, 2s and let A be a compact subset of the vertical segment tpe 1 , sq P R 2nˆR " H n : s P r´1{4, 1{4su, such that dim H n A " 2 dim E A " α, where e j is the j-th standard basis vector in Euclidean space. Let U " tV P G h pn, mq : |ωpe 1 , wq| ą 1{2 for some w P V with |w| ď 1u.
Then U is a nonempty open set, and so µ n,m pUq ą 0. For pe 1 , sq, pe 1 , tq P A and V P U, there exists w P V with |w| ď 2|s´t| such that ωpe 1 , wq " s´t. Hence It follows that This shows that (3.24) is sharp for dim H n A ď 2, which by (3.29) and Lemma 3.1, implies the sharpness of (3.23) for dim E A ď 1.
For an example with any Heisenberg dimension in the range p2, 2n`1´ms, let A be a set in H n with where C α Ď C n is a compact set of Euclidean dimension α P r0, 2n´1s and I is a compact interval of positive length (this is based on Case 2 from [4, Section 4.1]; see also Appendix B for an explicit construction). Then for all V P G h pn, mq. By varying α and using Lemma 3.3, this shows that (3.24) is sharp if 2 ď dim H n A ď 2n`1´m. By the Dimension Comparison Principle (see (3.29)), this implies that the lower bound of (3.23) is sharp for dim E A ď 2n´m.

An improved bound in H
In this section we prove a result for Euclidean dimension distortion under projections in H " H 1 , which for n " 1 improves Theorem 1.2 in a small range.
Since the family of nontrivial horizontal subgroups in H is the one dimensional family of lines in Cˆt0u through the origin, the symbols V θ and V θ will be used for the 1-dimensional subspaces containing pcos θ, sin θq and pcos θ, sin θ, 0q, respectively. In this section, the notation P V K θ will indicate either P L ; the left/right designation will only be used if necessary. The proofs of Lemmas 4.1 and 4.2 use the right coset formula in computations, but by symmetry this is inessential.
The following (standard) lemma essentially says that the family of vertical projections obeys a weak version of transversality with respect to the Euclidean metric.
The following lemma is the main result of this section, which will be converted to a projection theorem via a standard technique. The proof will only be sketched since it is similar to the case of Euclidean projections in R 3 [25], and also to the Korányi metric case of left projections in H [15]; the main emphasis will be on the steps which differ from [15]. For any κ ą 2ps´1q 3 , there exist δ 0 , η ą 0 such that for all δ P p0, δ 0 q.
Proof. Assume without loss of generality that ν is supported in the unit ball, and that κ ă s´1. Choose η with where the right hand side is positive by assumption. Define A AE B to mean A À δ´O pηq B, and write A « B if A AE B and B AE A.
Let Z be the set of x's occurring in (4.33). The argument that follows works for any δ ą 0 sufficiently small, so we assume δ 0 has been suitably chosen and δ P p0, δ 0 q. For any such δ, dyadic pigeonholing gives a set Z 1 Ď Z with νpZ 1 q « νpZq and a fixed dyadic number t with δ ď t À 1, such that for each x P Z 1 there are three sets H 1 pxq, H 2 pxq, H 3 pxq Ď r0, πq that are « 1-separated for each x, each with H 1 -measure « 1, such that 2tq is the Euclidean annulus around x of inner radius t and outer radius 2t. This pigeonholing step is virtually identical to those in [25] and [15] (where more details are provided). Let where x " pz, τ q and ℓpz, wq is the line through z and w in R 2 . The lemma will follow from the outer two parts of (4.37) where x " i x i means that (4.38) t ď d E px, x i q ă 2t and d E´PV K θ pxq, P V K θ px i q¯ă δ, for some angle θ P H i pxq.
The lower bound of (4.37) essentially follows by fixing x P Z 1 , establishing the lower bound tδ s´κ´1 on the ν-measure of the set of x i 's satisfying x " i x i , integrating over x 1 , x 2 and x 3 to get t 3 δ 3ps´κ´1q , integrating over x P Z 1 and using νpZq « νpZ 1 q. This argument is similar to the one in [25], except that here as in [15] the points px, x 1 , x 2 , x 3 q have the additional requirement that they must be in Λ. For the lower bound tδ s´κ´1 on the ν-measure of the set of x i 's satisfying x " i x i , the proof proceeds by sorting the points x i according to the interval I k of length δ{t containing the corresponding angle θ in (4.38), using (4.35) to bound the contribution of these points below by δ s´κ and then adding up « tδ´1 such intervals (this is where Lemma 4.1 is needed to ensure disjointness).
If t AE δ α this proves the lower bound of (4.37). If t Ç δ α , then to adjust this argument to accommodate the requirement that px, x 1 , x 2 , x 3 q P Λ, group the intervals I k of length δ{t into larger intervals J j of length δ α {t, so that each group contributes « δ α´1 δ s´κ to the lower bound. It suffices to show that for fixed x, x 1 , x 3 with |z´z 1 |, |z´z 3 | ě t{2, and fixed j, the set E :" tx 2 " pz 2 , t 2 q P H : d E pz 2 , ℓpz 1 , z 3 qq ă δ α , (4.38) holds for some θ P H 2 pxq X J j u is contained in a Euclidean ball of radius « δ α ; the excision of this set will therefore not harm the lower bound of δ α´1`s´κ since δ αs is much smaller than δ α´1`s´κ , by the definition of α in (4.36) (provided the Opηq factor is chosen sufficiently large).
To see that E is contained in a ball of radius « δ α , fix some x 2 " pz 2 , τ 2 q P E. The projection of E down to R 2ˆt 0u will be shown to be contained in where N refers to Euclidean neighbourhood. The first set in the intersection comes from the definition of E. For the second set, by (4.38) the line ℓpz, z 2 q is at an angle of θ to the x-axis (up to an error À δ{t), where θ is the angle from (4.38). Since by definition of E the corresponding angles of all other points in E have been grouped into one interval of length δ α {t, all other lines ℓpz, z 1 2 q with x 1 2 P E are within an angle À δ α {t of the line ℓpz, z 2 q. Since by (4.38) all points z 1 2 P E satisfy |z 1 2´z | ď 2t, it follows that the part of E in all of these lines is contained in N Cδ α pℓpz, z 2 qq for some large enough constant C. This proves the projection of E down to R 2ˆt 0u is contained in the set in (4.39). The set in (4.39) is contained in a ball of radius « δ α ; this follows from δ Opηq -transversality of the lines ℓpz 1 , z 3 q and ℓpz, z 2 q. This transversality is a simple geometric consequence of the angle separation assumption on the sets H i pxq; an explicit proof is given in [15]. It remains to bound the distances between the last coordinate. By (4.38) and the preceding argument, any two points pz 2 , τ 2 q and pz 1 2 , τ 1 2 q in E satisfy (4.40) |z 2´z for some θ and θ 1 . Combining the second and third parts of (4.40) with (4.41) and (4.42) respectively yields (4.43) τ´τ 2´1 2 ωpz, z 2 q À δ, τ´τ 1 2´1 2 ωpz, z 1 2 q À δ.
Combining this with the first part of (4.40) and using the triangle inequality gives |τ 2´τ which proves that E is contained in a Euclidean ball of radius « δ α , and finishes the proof of the lower bound of (4.37).
For the upper bound, the case t AE δ α follows by integrating over px, x 1 , x 2 , x 3 q and using the Frostman condition on ν, so assume that t Ç δ α . Let A " Apx 1 , x 2 , x 3 q " tx P Z 1 : px, x 1 , x 2 , x 3 q P Λ and x " i x i for all iu.
The upper bound in (4.37) will be shown by bounding νpAq and then integrating over px 1 , x 2 , x 3 q. Let where x " pz, τ q. By similar working to that used to show (4.43), As in the left projection case ( [15]), the Jacobian satisfies by the definition of Λ. Hence It follows that (4.44) νpA 1 q AE δ p1´αqs t´s.
To bound νpAq it remains to bound νpAzA 1 q. If x P AzA 1 then |z´z i | ă |t´t i |{10 for some i, so by the condition x " i x i and by similar working to (3.27), there exists θ such that Hence νpAzA 1 q À δ s . Combining with (4.44) gives νpAq AE max δ p1´αqs t´s, δ s ( AE δ p1´αqs t´s, since t À 1. Integrating over x 1 , x 2 , x 3 gives which is the upper bound of (4.37). If t AE δ α , then combining the lower and upper bounds of (4.37) gives Since s ą 1, this simplifies to νpZq AE δ 3αps´1q´3ps´κ´1q , and therefore νpZq ď δ η by the definition of α in (4.36). This finishes the proof if t AE δ α . Now assume t Ç δ α . In this case the lower and upper bounds of (4.37) give (4.45) νpZqt 3 δ 3ps´κ´1q AE δ p1´αqs t s .
Proof. The Euclidean part follows from [15,Lemma 2.1], which says that any result of the type in Lemma 4.2 implies a corresponding projection theorem with lower bound s´κ for sets of dimension s. The non-Euclidean part for right coset projections follows from the Euclidean bound, the dimension comparison principle and Lemma 3.1.

Open questions
Sharp Euclidean lower bounds. The Euclidean lower bound in Theorem 1.2 is probably not sharp in the entire range. So the first obvious way to further this work would be to improve this bound, ideally finding sharp dimension distortion bounds. Since the projection maps are now viewed as maps from R 2n`1 to R 2n´m`1 , purely Euclidean methods could in principle be applied to improve dimension distortion bounds. For instance, Fourier restriction methods used for example in [24] might lead to improvements. As we showed, when studying the problem as a Euclidean one, left and right coset projections cause the same dimension distortion. Therefore improving the bound in this direction could further improve the bound for the two problems relative to the more natural metrics that go with each one.
Sharp VzH n lower bounds. The method we employed here was to study the problem as a Euclidean one, and then apply the Dimension Comparison Principle to obtain dimension distortion bounds with respect to the more natural metric d VzH n . So our bounds are obtained considering the worst dimension distortion by projections and the worst dimension drop by dimension comparison. In principle, these two things need not happen simultaneously so better bounds could potentially be obtained by considering the maps P R V K as maps from pH n , d H n q to pV K , d VzH n q or pV K , d E q and estimating energy integrals with respect to these metrics directly.
Projections of subsets with specific structure. In [2] the authors gave evidence to their conjectured almost sure lower bound, in H, by exhibiting some subsets of H with specific structure that do adhere to their conjecture. For instance, if the set S is either a C 1 curve or a C 1 surface then dim H P L θ S ě dim H S for all but at most 2 values of θ. Does something similar hold in higher dimensions for the projections P L V K and/or P R V K ? Structure of pV K , d VzH n q. This problem was mentioned to in Section 2.2. The properties of this space discussed in that section hint that it might have the structure of a non-equiregular Carnot-Carathéodory space. So the problem is that of finding bracket generating vector fields in R 2n´m`1 such that R 2n´m`1 with the induced Carnot-Carathéodory distance is isometrically (or at least bi-Lipschitz) equivalent to pV K , d VzH n q. Such a description of the space may also lead to improvements in dimension distortion bounds by projections as it could provide a better understanding of the metric itself.

Appendix A. A slicing result
Let H m denote the m-dimensional Hausdorff measure on Euclidean space, with respect to the Euclidean metric. Let MpAq be the class of compactly supported, nonzero, finite Radon measures on a set A Ď H n . Let NpE, δq be the δ-neighbourhood of a set E Ď H n with respect to the Euclidean metric.
Theorem A.1 (A slicing result). Let A Ă H n be a Borel set such that dim A ą m`1 and 0 ă H dim E A A ă 8 for 1 ď m ď n. Then for µ n,m -almost every V P G h pn, mq, Note the great deal of similarity between this theorem and Theorem 1.5 in [3]. In fact the proof follows the same techniques and only differ in that here we consider Hausdorff dimension with respect to the Euclidean metric.
Proof. Eilenberg's inequality (Theorem 13.3.1 in [8]) tells us that for every V, Therefore, we only need to prove the dimension lower bound. For this, we will make use of sliced measures in the sense of [20] (Section 10.1). By Eq. (10.6) in [20], we know that for µ P MpAq there exists a family of measures µ V K v , defined for H m -a.e. v P V, each supported on V K v, such that for any non-negative continuous function ϕ compactly supported on H n and any Borel set B Ă V, the map v Þ Ñ ş V K v ϕpuq dµ V K v puq is Borel measurable and satisfies vq dH m pvq " µpAq ą 0, so that at least a H m -positive measure set of the measures µ V K v are in MpA X V K vq. Hence, we want to pick a measure µ P MpAq such that P V# µ ! H m for µ n,m -almost every V. As the next claim will show, this is possible precisely when dim E A ą m`1.
Claim: Let σ ą m`1 and assume µ P MpH n q satisfies µpB E pp, rqq ď r σ for all p P H n and r ą 0. Then P V# µ ! H m t V for µ n,m -almost every V.
To see this, denote by π : H n Ñ R 2n the bundle map πpz, tq " z, and note that P V# µpB V pv, rqq " µpP´1 V pB V pv, rqqq " µptp P H n : |P V ppq´v| ă ruq. Now, Theorem 2.12 in [22] tells us that P V# µ ! H m if and only if lim inf δÑ0 δ´mP V# µpB V pv, δqq ă 8 for P V# µ-almost every v P V.
Using Fatou's lemma, and Fubini (see e.g. Theorem 1.14 in [22]), we compute: ż where the last step follows from Lemma 2.4 in [3]. We now focus our attention on showing finiteness of this last integral. Since supppµq is compact, we can fix R ą 0 such that supppµq Ă B E p0, Rq. For z P R 2n the set tq P H n : |πpqq´z| ď ru is a cylinder with radius r, so tq P H n : |πpqq´z| ď ru X supppµq Ă B 2n E pz, rqˆr´R, Rs. This cylinder can be covered by at most rCr´1s balls of radius r, where C " Cpn, Rq is independent of z and r. It follows that µptq P H n : |πpqq´z| ď ruq À r σ´1 . Therefore, ż Since σ´1 ą m it follows that ş 8 1 r 1´σ m dr ă 8. This tells us that ż H n ż H n |πpqq´πppq|´m dµpqq dµppq À µpH n qˆµpH n q`ż By Frostman's lemma, if dim A " α ą m`1 with 0 ă H α pAq ă 8, then we may choose µ P MpAq to be a suitable restriction of H α such that µpB E pp, rqq ď r α for all p P H n and r ą 0. From the claim, we know P V# µ ! H m for µ n,m -almost every V P G h pn, mq. As noted before, it follows that for µ n,m -almost every V P G h pn, mq, the measure µ V K v is in MpA X V K vq for all v in a set of positive H m measure.
We now aim to show that if m`1 ă s ă α, then for µ n,m -almost every V P G h pn, mq, By Fatou's lemma, Tonelli's theorem, and by applying (A.46) with B " Bpv, δq and letting δ Ñ 0, we can compute: This last integral is not quite I s pµ, d E q, and in fact the singularity in the kernel |q| m´s |πpqq|´m is stronger than the one in the kernel |q|´s. Nevertheless, we will show this integral is finite following the same approach as in the proof of Theorem 1.5 in [3], by showing that the inner integral is finite for all p and using the fact that µpH n q ă 8.
If we denote by L´p the Euclidean left translation by´p, the inner integral can be written as ż H n |p´q| m´s |πppq´πpqq|´m dµpqq " ż H n |q| m´s |πpqq|´m dL´p # µpqq.
Moreover, it is clear that L´p # µpH n q " µpH n q and L´p # µpB E pq, rqq ď r s for every q P H n and r ą 0. Furthermore, since µ is compactly supported, by scaling we may assume the support of L´p # µ is contained in B E p0, 1q. Therefore it is enough to show that ż H n |q| m´s |πpqq|´m dµpqq À 1, whenever µ P MpB E p0, 1qq and satisfies µpB E pp, rqq ď r α for all p P H n and r ą 0. Writing q " pζ, τ q we have ż We look at these two quantities separately, the first one being the easier to bound. Indeed, using a change of variables and recalling our choice of s, To bound the second integral we first split the domain of integration: tpζ, τ q P B E p0, 1q : |ζ| ă |τ |u " 8 ď i"0 pζ, τ q P B E p0, 1q : 2´i´1|τ | ď |ζ| ă 2´i|τ | ( ": Note that for pζ, τ q P A i , |ζ|´1 " 2 i |τ |´1. Therefore, where A i,j " pζ, τ q P B E p0, 1q : 2´i´j´2 ď |ζ| ď 2´i´j, 2´j´1 ď |τ | ă 2´j ( . To estimate µpA i,j q we see that A i,j Ă B 2n E p0, 2´i´jqˆr´2´j, 2´js. Hence, there exists a constant C ą 0 independent of i and j such that A i,j can be covered by at most C 2´j 2´i´j " C2 i balls of radius 2´i´j. The Frostman condition on µ now tells us that µpA i,j q À 2 i 2´α pi`j q. Going back to the sum we are trying to bound, we get 8 ÿ i,j"0 2 im`js µpA i,j q À 8 ÿ i,j"0 2 ipm`1´αq`jps´αq , which is finite since m`1´α and s´α are both negative. Now to complete the proof of the proposition, for V P G h pn, mq write E V :" tv P V : µ V K v pH n q ą 0u, so that for v P E V , µ V K v P MpA X pV K vqq. Since, by the claim, we know P V# µ ! H m , equality in (A.46) with B " V tells us that H m pE V q ą 0. Furthermore, by the previous computation it follows that if m`1 ă s ă α then for µ n,m -almost every V P G h pn, mq, the energy I s´m pµ V K v , d E q is finite for H m -almost every v P E V . This tell us that dim E rA X pV K vqs ě s´m. Since E V is independent of s and α, the theorem follows by letting s Ñ α.
In the first case, let C β be a Cantor set in R with finite, nonzero β-dimensional Euclidean Hausdorff measure, and let A 1 " pC βˆt 0uqˆC jˆt 0u 2pn´1q´2jˆR , which clearly satisfies dim E A 1 " α`1. Using H a b to denote the a-dimensional Euclidean Hausdorff measure living on R b , the measure µ :"´H β 2ˆH 2j 2pn´1qˆH 1 1¯| A 1 , is nonzero and supported on A 1 , and satisfies µ pB H n ppz, tq, rqq À r α`2 , for all pz, tq P A 1 and 0 ă r ă 1. This can be proved similarly to the proof that the Hausdorff dimension of H n is 2n`2; using invariance by left translation and that the coordinates from C βˆt 0u vanish in the symplectic form. Hence dim H n pA 1 q ě α`2, and therefore dim H n pA 1 q " α`2 by dimension comparison. By using homogeneous dilations, this implies that the set A :" pC βˆt 0uqˆr0, 1s 2jˆt 0u 2pn´1q´2jˆr 0, 1s. satisfies (B.48). This finishes the construction in the case of (B.49). The odd case in (B.50) is similar, except that A is defined by A "´C1`β 2ˆt 0u¯2ˆr0, 1s 2jˆt 0u 2pn´2q´2jˆr 0, 1s.