THE EFFECT OF PROJECTIONS ON DIMENSION IN THE HEISENBERG GROUP

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1 THE EFFECT OF PROJECTIONS ON DIMENSION IN THE HEISENBERG GROUP ZOLTÁN M. BLOGH, ESTIBLITZ DURND CRTGEN, KTRIN FÄSSLER, PERTTI MTTIL, ND JEREMY T. TYSON bstract. We prove analogs of classical almost sure dimension theorems for Euclidean projection mappings in the first Heisenberg group, equipped with a sub-riemannian metric. Contents 1. Introduction 1. Review of background material 6 3. Projections onto horizontal subspaces 9 4. Universal bounds for vertical projections lmost sure bounds for vertical projections Projections of submanifolds 6 7. Projections of subsets of horizontal or vertical planes 8 8. Final remarks 4 References 4 1. Introduction In this paper, we study projection mappings from the Heisenberg group onto horizontal lines and complementary vertical planes. In particular we consider the effect of such mappings on the Hausdorff dimensions and Hausdorff measure of subsets of the Heisenberg group considered with respect to a sub-riemannian metric. Our results are analogs, in sub-riemannian geometry, for classical theorems of Marstrand [15]. We shall employ potential theoretic methods first used in this context by Kaufman in [1] and later generalized in [16]. There have been many studies on Marstrand type projection results. For example, a general Fourier analytic machinery for projection-type theorems was developed by Peres and Schlag in []. See also the survey [18] for an overview of the subject. This paper represents part of an extensive program aimed at developing geometric measure theory beyond the Euclidean setting, the origins of which date back to Gromov s groundbreaking treatise [1]. The Heisenberg group H is the unique analytic nilpotent Lie group whose background manifold is R 3 and whose Lie algebra h admits a vector space decomposition h = v 1 v, Date: May 4, 11. ZMB and KF supported by the Swiss National Science Foundation and European Science Foundation Project HC. EDC supported in part by DGES (Spain) Project MTM and by Grant P6-6. PM supported by cademy of Finland grants and , JTT supported by NSF Grant DMS

2 where v 1 has dimension two, v has dimension one, and the Lie bracket identities [v 1, v 1 ] = v, [h, v ] = hold. We identify H with C R = R 3 through exponential coordinates. Points in H are denoted p = (z, t). We work throughout this paper with the following convention for the group law: (1.1) (z, t) (ζ, τ) = (z + ζ, t + τ + Im(z ζ)). Our results are formulated with respect to a sub-riemannian structure on the Heisenberg group. We will work primarily with the well known Heisenberg metric on H (also known as the Korányi metric). This is the left invariant metric given by where H is the gauge norm defined by d H (p, q) = q 1 p H, p H = ( z 4 + t ) 1/4. Note that d H is bi-lipschitz equivalent to the Carnot-Carathéodory metric on H which can be defined using horizontal curves. n absolutely continuous curve γ : I H R 3 on an interval I in R is called horizontal if γ(s) H γ(s) H for almost every s I, where H p H = span{x p, Y p } with X = x + y t and Y = y x t. ll results which we shall obtain regarding Hausdorff dimensions of subsets of H are unchanged under bi-lipschitz change of the metric. The advantage of working with the metric d H, rather than using the Carnot-Carathéodory metric, is its simple and explicit form. There is also a one-parameter family of nonisotropic dilation mappings (δ r ) r>, given by δ r (z, t) = (rz, r t). We recall that the Hausdorff dimension of the metric space (H, d H ) is equal to 4. In fact, (H, d H ) is an hlfors 4-regular metric space. The Heisenberg group H has the structure of an R bundle over the plane R. We write π : H R for the mapping π(z, t) = z and note that π is 1-Lipschitz as a map from (H, d H ) to (R, d E ). Here and throughout this paper, d E denotes the Euclidean metric on any Euclidean space. subgroup G of H is called a homogeneous subgroup if it is invariant under the dilation semigroup (δ r ) r>, i.e., p G, r > δ r (p) G. Observe that under the aforementioned identification of H with R 3 homogeneous subgroups of H are vector subspaces of R 3. For fixed [, π), let V be the one-dimensional subspace of R 3 spanned by the vector (e i, ). Then V is a homogeneous subgroup of H. Let W be the Euclidean orthogonal complement of V, i.e., the two-dimensional subspace of R 3 spanned by the vectors (ie i, ) and (, 1). Then W is also a homogeneous subgroup of H. We will identify V with R via the global chart (1.) (re i, ) ϕ V r, and we will identify W with R via the global chart (1.3) (aie i, t) ϕ W (a, t).

3 In this paper, we call the homogeneous subgroups V, [, π), horizontal subgroups and we call the subgroups W, [, π), vertical subgroups. Both types of subgroups are abelian subgroups of H, in addition, vertical subgroups are normal subgroups of H. Note also that the restriction of d H into a horizontal subgroup V coincides with the restriction of the Euclidean metric of R 3 to V. We therefore may speak about metric properties of the horizontal subgroups V without reference to the metric. On the other hand, the restriction of d H into a vertical subgroup W is given by (1.4) d H (ϕ 1 W (a, t), ϕ 1 W (a, t )) = ( (a a ) 4 + (t t ) ) 1/4 and is comparable to the parabolic (heat) metric a a + t t 1/ on R. For each, the pair V and W induces a semidirect group splitting H = W V. For p H, we write p = p W p V where p W W and p V V. In this way, we define the horizontal projection p V : H V and vertical projection p W : H W by the formulas and p V (p) = p V p W (p) = p W. Explicit expressions for these mappings appear in (.6) and (.7). The semidirect splitting of H (and more general Carnot groups) into horizontal and vertical subgroups has played a key role in recent developments concerning intrinsic sub-riemannian submanifold geometry and sub-riemannian geometric measure theory, see for example [9], [8], [14], and [19]. The mappings p V and p W have rather different character. The horizontal projection maps p V are linear projection maps with respect to the underlying Euclidean structure on R 3, moreover, they are also Lipschitz maps (with Lipschitz constant 1) and homogeneous group homomorphisms of H. On the other hand, the vertical projection mappings p W are neither linear, nor (Euclidean) projections, nor group homomorphisms. These facts highlight the difficulty of working with the vertical projection mappings in the Heisenberg group. Nevertheless, we will ultimately be able to derive estimates for the effect of vertical projection on the Hausdorff dimensions of sets. We denote by dim the Hausdorff dimension in a general metric space, and by H s, s >, the corresponding family of Hausdorff measures. By Hδ s, δ >, we denote the Hausdorff premeasures in dimension s. We will work with these notions for both the Heisenberg and Euclidean metrics d H and d E on H R 3, so we will take care to specify the metric with which we are working, writing HH s, Hs E and dim H, dim E. Similarly we will denote by B E (p, r), resp. B H (p, r), the ball of radius r and center p in the metric space (R 3, d E ), resp. (H, d H ). We emphasize that we always consider closed balls in this paper. Our main theorems provide universal and almost sure estimates for the (Heisenberg) dimensions of horizontal and vertical projections of Borel subsets of H. By a universal estimate we mean an inequality relating either dim H p V () or dim H p W () to dim H which is valid for all sets and all angles. By an almost sure estimate we mean an inequality relating these quantities which is valid for all sets and for L 1 -almost every angle. Henceforth all measure theoretic statements involving the angle will be done with respect to the Lebesgue measure L 1. 3

4 Let H be a Borel set. Since the horizontal projection maps are Lipschitz and the horizontal subspaces are 1-dimensional, the estimate (1.5) dim p V () min{1, dim H } holds for all. Note that the dimension of p V () with respect to the Heisenberg metric is the same as with respect to the Euclidean distance. We first state which universal and almost sure lower bounds hold for the dimensions of horizontal projections. Theorem 1.1 (Universal lower bounds for horizontal projections). Let H be a Borel set. Then (1.6) dim p V () max{, dim H 3} for all. The estimate in (1.6) is sharp. Theorem 1. (lmost sure lower bounds for horizontal projections). Let H be a Borel set. Then (1.7) dim p V () max{, min{dim H, 1}} for a.e.. If dim H > 3, then H 1 (p V ()) > for a.e.. The estimate in (1.7) is sharp. Figure 1 illustrates the sets of universal and almost sure dimension pairs for horizontal projections on H. dim p () H V dim p () H V dim () H dim () H Figure 1. (a) Universal dimension pairs for horizontal projections; (b) almost sure dimension pairs for horizontal projections The proof of Theorem 1. is rather straightforward. It uses simple estimates for the dimension of the projection π() combined with classical almost sure dimension theorems for Euclidean projections. The sharpness parts of Theorem 1. and Theorem 1.1 are contained in Proposition 3.. The state of our knowledge regarding the effect of the vertical projections on Hausdorff dimension is less well advanced. However, we are able to obtain some results. Namely, we can show the following theorems. Note that the Hausdorff dimension of the vertical subgroups W with respect to the Heisenberg metric d H on H, is equal to 3. Theorem 1.3 (Universal upper and lower bounds for vertical projections). Let H be a Borel set. Then (1.8) dim H p W () min{ dim H, 1 (dim H + 3), 3} for all and (1.9) dim H p W () max {, 1 (dim H 1), dim H 5 } for all. The universal estimates in (1.8) and (1.9) are sharp. 4

5 Theorem 1.4 (lmost sure lower bounds for vertical projections). Let H be a Borel set. If dim H 1, then (1.1) dim H p W () dim H for a.e.. Consequently, for any, (1.11) dim H p W () max{min{dim H, 1}, dim H 5} for a.e.. The estimate (1.11) is sharp when dim H 1. dim p () H W dim p () H W dim () H dim () H Figure. (a) Universal dimension pairs for vertical projections; (b) almost sure dimension pairs for vertical projections (including conjectured sharp lower bound) The sharpness statement of Theorem 1.3 is discussed in Proposition 4.1. The upper bound (1.8) is also sharp as an almost sure statement, see Proposition 5.3. Examples which prove the sharpness of the lower bound (1.11) in Theorem 1.4 in the case when dim H 1 are given by subsets of the t-axis. We do not know whether the lower bound (1.11) is sharp in the case when 1 < dim H < 4 but we suspect not. We formulate the following Conjecture 1.5. For all H, dim H p W () min{dim H, 3} for a.e.. If dim H > 3, then H 3 H (p W ()) > for a.e.. Proposition 6.1 and Theorem 6.3 provide partial evidence in support of Conjecture 1.5. Figure illustrates the sets of universal and almost sure dimension pairs for vertical projections on H (including the conjectured sharp lower bound). The lower bounds in Theorem 1.4 can be improved in case the set is a subset of either a horizontal plane or a vertical plane. See Section 7 for details. We would like to emphasize an important difference between Theorems 1. and 1.4 and their Euclidean predecessor, see Theorem.3 below. Namely, for any Borel set R n, the almost sure dimension of the image P V () under a Euclidean projection on an m-dimensional subspace V can be computed exactly as a function of dim E and m. No similar formula holds in the Heisenberg setting, at least for arbitrary Borel sets. Indeed, the best result which can be obtained is a pair of (distinct) upper and lower bounds for the Heisenberg dimensions of the projections. We give a variety of examples to demonstrate the sharpness of our estimates. Finally, let us remark that we do obtain an exact formula for the L 1 -almost sure dimension of 5

6 the horizontal projection in the low codimensional case dim H > 3. Conjecturally, a similar exact formula holds for the vertical projections under the same assumption on dim H. We conclude this introduction with an outline of the paper. In Section we recall preliminary information concerning almost sure dimension theorems in Euclidean space and the dimension comparison principle in the Heisenberg group. Section 3 treats the case of the horizontal projection mappings and contains the proof of Theorems 1.1 and 1.. Section 4 contains the proof of the universal dimension bounds for the vertical projection mappings: Theorem 1.3. The main results of the paper concerning the almost sure dimension theorem for vertical projections, Theorem 1.4 and the related examples, are presented in Section 5. Since our results on almost sure dimensions of vertical projections are rather incomplete, we will discuss several classes of examples where we have a better understanding of the behavior of the dimension of the projections. The first such class consists of sets with a certain degree of regularity. This is discussed in Section 6. In Section 7 we sharpen the analysis of the vertical projections, obtaining improved dimension estimates for projections of subsets of horizontal or vertical planes. Section 8 contains remarks and open questions motivated by this study cknowledgements. Research for this paper was initiated while EDC and JTT were guests in the Mathematics Institute of the University of Bern in Fall 9 and completed while PM was a guest of the same institute in Fall 1. The hospitality of the institute is gratefully appreciated.. Review of background material.1. Dimension and Euclidean projections. Theorems 1. and 1.4 are adaptations to the Heisenberg setting of classical almost sure dimension theorems for Euclidean projections, proved by Marstrand in the plane [15] and generalized in [16]. We briefly recall the Euclidean theorems. Definition.1. Let m and n be integers with < m < n. The Grassmanian G(n, m) is the space of all m-dimensional linear subspaces of R n. It is possible to introduce a natural measure γ n,m on G(n, m). In the case m = 1 this measure is fairly simple to describe. In fact, the Grassmanian G(n, 1) coincides with the real projective space P n 1 R, and the measure in question is the pushforward of the surface measure from S n 1 under the canonical quotient map S n 1 P n 1 R. For instance, G(, 1) can be identified with PR 1, or even more explicitly with the interval [, π) (by identifying a line through the origin in R with the angle [, π) which it makes with the positive x-axis). Under the latter identification, the measure in question is just d. Via the canonical identification of the Grassmanians G(n, m) and G(n, n m), we could also describe the natural measure on G(n, n 1) quite explicitly. However, for m n the story is more complicated. We refer to [17, 3] for the construction of the measure γ n,m on G(n, m). It can be checked that γ n,m is equivariant with respect to the usual action of the orthogonal group O(n) on G(n, m). Remark.. The measure γ n,m can be constructed in another manner. The Grassmanian G(n, m) is a smooth manifold of dimension m(n m), and is also a metric space when equipped with the distance function d(v, W ) = P V P W. Here P V : R n V denotes orthogonal projection from R n onto a subspace V, and denotes the operator norm. Up to 6

7 a multiplicative constant, the measure γ n,m coincides with the Hausdorff measure H m(n m) on the metric space (G(n, m), d). This follows easily from the fact that both of the measures in question are O(n) equivariant, and hence uniformly distributed. See Definition 3.3 and Theorem 3.4 in [17] for additional details. Theorem.3 (Euclidean Projection Theorem). Let m and n be integers with < m < n and let R n be a Borel set. If dim E m, then dim E P V () = dim E for γ n,m - a.e. V G(n, m). If dim E > m, then H m (P V ()) > for γ n,m -a.e. V G(n, m). In particular, (.1) dim E P V () = min{dim E, m} for γ n,m -a.e. V. Suslin set is the continuous image of a Borel set. Theorem.3 extends to Suslin sets. Frostman s lemma is a standard tool used in the proof of lower bounds for Hausdorff dimension. We denote by M() the collection of positive, finite Borel regular measures supported on a set of a metric space X. Theorem.4 (Frostman s lemma). Let be a Borel (Suslin) subset of a complete metric space (X, d). Suppose that there exists s >, µ M(), and r (, ] so that the inequality (.) µ(b(x, r)) r s holds for all x and < r < r. Then H s () >. In particular, dim s. Conversely, if H s () > then there exists a measure µ M() so that (.) holds for all x and r >. See, e.g., [7, Proposition 4.], [11] or [17, Theorem 8.17]. We say that µ satisfies an upper mass bound on with exponent s if (.) holds for all x and < r < r. Next, we state the energy version of Frostman s lemma. This follows easily from Theorem.4, see [17, Chapter 8]. Definition.5. Let (X, d) be a metric space and let µ M(X). For s >, the s-energy of µ is I s (µ) = d(x, y) s dµ(x) dµ(y). X X Theorem.6 (Frostman s lemma, energy version). Let be a Borel (Suslin) subset of a complete metric space (X, d) and let s > be such that there exists µ M() with I s (µ) <. Then dim s. Conversely, if is a Borel (Suslin) subset of a complete metric space (X, d) and s < dim, then there exists µ M() with I s (µ) <... Dimension comparison principle. We will make use of the recent solution to the dimension comparison problem in H. This problem, originally posed by Gromov [1,.6.C], asks for sharp estimates relating the Euclidean and Heisenberg measures and dimensions of subsets of H. nearly complete answer was given by Balogh Rickly Serra-Cassano [3]; the story was completed by Balogh Tyson [4] who gave examples demonstrating the sharpness of the lower bound. We state the final result, in its sharp form. Theorem.7 (Dimension comparison in the Heisenberg group). Let H be a Borel set with dim E = α [, 3] and dim H = β [, 4]. Then (.3) max{α, α } =: β (α) β β + (α) := min{α, α + 1}. 7

8 Furthermore, for any pair (α, β) [, 3] [, 4] satisfying β (α) β β + (α), there exists a compact set α,β H with dim E α,β = α and dim H α,β = β. Theorem.7 was generalized to arbitrary Carnot groups by Balogh, Tyson and Warhurst [5]. From now on, we refer to the estimates in (.3) as the dimension comparison principle for the Heisenberg group H. We will also use the dimension comparison principle in vertical subgroups of H. Due to the special form (1.4) of the restriction of the Korányi metric to such subspaces, we obtain stronger dimension comparison estimates therein. To wit, we have Theorem.8 (Dimension comparison in vertical subgroups of the Heisenberg group). Let W be a Borel set contained in some vertical subgroup W H, with dim E = α [, ] and dim H = β [, 3]. Then (.4) max{α, α 1} =: β W (α) β β W + (α) := min{α, α + 1}. Theorem.8 can be proved by adapting the arguments from [5]..3. Explicit formulas for horizontal and vertical projections. We present explicit formulas for the projection mappings p V and p W, and for the distance between points in H and the corresponding distance between their projections. Such formulas will be useful in the proofs of Theorems 1. and 1.4. Let [, π) and let p = (z, t) H. We recall that the projections p V and p W are determined by the identity (.5) p = p W p V. The horizontal projection p V coincides with the Euclidean projection P V : R 3 V and is given by (.6) p V (z, t) = p V = ( Re(e i z)e i, ). The vertical projection p W can then be determined via (.5) and is given by (.7) p W (z, t) = p W = ( Im(e i z)ie i, t Im(e i z ) ). Denote by p = (z, t) and q = (ζ, τ) two points in H. Observing that Im((z ζ)(z + ζ)) = Im(zζ) and using the formula (1.1) for the group law in H, we record the following expression for the distance between p and q: (.8) d 4 H(p, q) = q 1 p 4 H = z ζ 4 + (t τ + z ζ sin(ϕ 1 ϕ )). Here we wrote ϕ 1 = arg(z ζ) and ϕ = arg(z + ζ). Similarly, the distance between p V (p) and p V (q) can be expressed in the form d H (p V (p), p V (q)) = p V (q) 1 p V (p) H = z ζ cos(ϕ 1 ). Finally, the distance between p W (p) and p W (q) can be expressed in the form (.9) d 4 H(p W (p), p W (q)) = p W (q) 1 p W (p) 4 H = z ζ 4 sin 4 (ϕ 1 ) + (t τ z ζ sin(ϕ + ϕ 1 )). Note that the vertical projections p W : H W are locally 1 -Hölder continuous with respect to the Heisenberg metric. This is an easy computation involving the explicit formula for the projection. 8

9 3. Projections onto horizontal subspaces In this section, we discuss the effect of horizontal projections on Borel sets in the Heisenberg group. We begin with a lemma on the relationship between the Heisenberg dimension of a set in H and the Euclidean dimension of its planar projection. Lemma 3.1. For any set H, we have dim E π() dim H. Proof. We may assume without loss of generality that is bounded. In fact, let us assume that t 1 for all points p = (z, t). Let s > dim E π(), let ɛ >, and cover the set π() with a family of Euclidean balls {B E (z i, r i )} i so that i rs i < ɛ. Since π : (H, d H ) (R, d E ) is 1-Lipschitz, the fiber π 1 (B E (z, r)) contains the ball B H ((z, t), r) for any t R. We can choose an absolute constant C > and N i C r i values t ij so that the family {B H ((z i, t ij ), C r i )} j covers the set B E (z i, r i ) [ 1, 1]. Then {B H ((z i, t ij ), C r i )} i,j covers the set. Denoting by rad(b) the radius of a ball B, we compute rad(b H ((z i, t ij ), C r i )) s+ = N i (C r i ) s+ C s+3 ri s C s+3 ɛ. i,j i i Letting ɛ gives H s+ H () = so dim H s +. Letting s dim E π() completes the proof. Proof of Theorems 1.1 and 1.. Let H satisfy dim H >. By Lemma 3.1, dim E π() dim H. Let us identify the one-dimensional subspace of R spanned by the vector e i with the corresponding one-dimensional subspace V H. This allows us to consider the Euclidean projection map P V as a (1-Lipschitz) map from R to V. pplying the Euclidean Projection Theorem.3 to π() (note that π() is a Suslin set) and noting that p V = P V π, we find dim p V () = dim E P V (π()) min{1, dim E π()} min{1, dim H } for a.e.. This proves (1.7). In the case when dim H > 3, we use the second part of Theorem.3 to arrive at the desired conclusion H 1 (p V ()) > for a.e.. Finally, for any, we have The proof is complete. dim H p V = dim E P V (π()) dim E π() 1 dim H 3. Both the universal bounds and the almost sure bounds for dimension distortion by horizontal projections are sharp. We collect relevant examples demonstrating this in the following proposition. Proposition 3.. In each of the following statements, the set is a compact subset of H. (a) For all β 1 there exists so that dim H = β and dim p V () = β for all. (b) For all 1 β 4 there exists so that dim H = β and dim p V () = 1 for all. (c) For all β 3 there exists so that dim H = β and dim p V () =. If β we can choose so that dim p V () = for all. (d) For all 3 β 4 there exists so that dim H = β and dim p V () = β 3. (e) For all β 3 there exists so that dim H = β and dim p V () = β for all. 9

10 Recall that a Borel set E is called an s-set, for some s, if < H s (E) <. bounded metric space (X, d) is said to be hlfors regular of dimension s if there exists a measure µ M(X) and a constant C 1 so that C 1 r s µ(b(x, r)) Cr s for all x X and < r < diam X. If (X, d) is hlfors regular of dimension s, then dim X = s and µ is comparable to the Hausdorff measure H s. Proof. For part (a), let V and π/ V π/ be compact β-sets. The set = π/ verifies the stated conditions. To show part (b) it suffices to construct a compact set with dim H = β and such that π() is a planar set which projects onto a 1-dimensional subset of V for every. We consider two cases. First, assume that 1 β 3. Let S [, 1] be a compact (β 1)/-set and let = π, where = {(x, t) : x 1, t S} and π = {(iy, t) : y 1, t S}. Since the restriction of d H to any vertical subgroup is comparable with the heat metric, dim H = 1 + dim S = β. The set π() is the union of two line segments which form a right angle at the origin. For = and = π one of the two segments is projected to a single point, but the projection of the entire set π() on V is 1-dimensional for every direction as desired. Next, assume that 3 < β 4. In this case, take the set to be the union of any compact set of Heisenberg Hausdorff dimension β with the set {(z, ) : z 1}. This completes the proof for part (b). We now turn to the proof of part (c). For the first claim, any compact β-set W suffices. In case β we can choose this compact set to be a subset of the t-axis, in which case p V () = {(, )} for all. Next we consider part (d). We may assume that 3 < β < 4. Let S R be a compact set which is hlfors regular of dimension (β 3), let B = {(iy, t) : y 1, t 1} and let = {p (x, ) : p B, x S}. Then p V () = {(x, ) : x S} has dimension β 3. It suffices to prove that dim H β. We will use Theorem.4. It suffices to show that the measure µ(e) := HH({p 3 (x, ) : p B } E) dh β 3 E (x) S has the upper mass bound (.) on with exponent β. Let B H (p, r) be a ball in (H, d H ) centered at p = p (x, ) with radius r. For x S, denote by B x the set of points of the form q (x, ), q B. Lemma 3.3. If B H (p, r) B x, then x x r and H 3 H (B H(p, r) B x ) Cr 3 for a constant C independent of p, r and x. ssuming the lemma we complete the proof in this case: µ(b H (p, r)) Cr 3 H β 3 E ([x r, x + r] ) C r β. Hence µ satisfies the upper mass bound (.) on with exponent β. By Theorem.4 dim H β. It remains to prove the lemma. Suppose that B H (p, r) B x. Then B H (p, r) B H (q, r) for some q B H (p, r) B x and H 3 H (B H(p, r) B x ) H 3 H (B H(q, r) B x ) Cr 3 since B x lies in a vertical plane in H. Furthermore, if q = (iy, t) (x, ) then x x r. This completes the proof of Lemma 3.3 and hence completes the construction for part (d). 1

11 Finally, we consider part (e). It suffices to construct a compact set H with dim H = β such that π() is a (β )-dimensional set in the plane whose dimension is preserved under P V for every. Let S R be a compact (β )-set and let = π, where = {(x, t) : x S, t 1} and π = {(iy, t) : y S, t 1}. Since the restriction of d H to any vertical subgroup is comparable with the heat metric, dim H = dim S + = β. Moreover, dim E π() = dim E S = β. The dimension of the set π( ) is preserved under P V except for = π, in which case π( ) is projected to a single point. n analogous statement holds for π( π ) and the exceptional direction =. ltogether, it follows that dim p V () = dim P V (π()) = β for all. The proof of Proposition 3. is complete. 4. Universal bounds for vertical projections In this section, we start to discuss the effect of vertical projections on the dimensions of Borel sets in the Heisenberg group. Our purpose is to prove Theorem 1.3. We recall that the projection map p W from H to the vertical subgroup W is given by (4.1) p W (z, t) = ( Im(e i z)ie i, t Im(e i z ) ). In contrast with the Euclidean case, this map is not Lipschitz continuous and hence does not a priori decrease dimension. Indeed, there are cases when this map increases dimension. Yet, there is still a certain control on the upper dimension bound coming from the local 1 -Hölder continuity of p W with respect to d H. Thus, for an arbitrary subset of H and for all, we have (4.) dim H p W () dim H. Example 4.1. Let = {(1 + i)s, ) : s [, 1]} H be a one-dimensional horizontal line segment. The projection of by the map p W is the graph of a parabola contained in the vertical subspace W. Thus p W () is a non-horizontal smooth curve and so has Hausdorff dimension equal to two. This shows that for 1-dimensional sets the upper bound (4.) cannot be improved. The proof of Theorem 1.3 is given in a series of propositions. Our first statement indicates the universal upper bounds which hold for the dimensions of vertical projections. Within a certain dimension range, the trivial upper bound given in (4.) can be improved. Proposition 4.. Let H be any Borel set. Then (4.3) dim H p W () min{ dim H, 1 (dim H + 3), 3} for every. The cases 3 < dim H 4 and dim H < 1 are trivial. The latter follows from the local 1 -Hölder continuity of p W. We will focus on the remaining case 1 dim H 3. The proof in this situation is more involved and uses a covering argument. Proposition 4.3. Let be a Borel subset of H with dim H [1, 3]. For all [, π) we have (4.4) dim H p W () 1 (dim H + 3). 11

12 The proof of this proposition is based on two preliminary results. Lemma 4.5, which describes the image of Heisenberg balls under vertical projections, and Lemma 4.6, which explains how this set can be covered efficiently by balls in the vertical plane. This allows us to find good covers for p W () which then yields the desired upper bound for the Hausdorff dimension. If not otherwise mentioned, we will in the following always identify the vertical plane W with R as described in (1.3). point p = (αie i, τ) in W will be written in coordinates as (α, τ). Let < r < 1 and x R. First, we describe the vertical projection in direction [, π) of a ball B H (p, r) with center p = (x, ) on the x-axis. We prove that there is a core curve γ x,r such that the image of the ball under p W lies in a small Euclidean neighborhood of the projected curve. For the following steps of the proof it will be essential to control the size of this neighborhood independently of the direction. This can be achieved if one uses a different curve depending on whether is close to π, or it is close to or π. Definition 4.4. The core curve γ x,r by γ x,r := related to x R and < r < 1 is a subset of H, given {(x + iy, x y) : y [ r, r]}, if [, π 4 ] [ 3π 4, π), {(x + x, ) : x [ r, r]}, if ( π 4, 3π 4 ). direct computation shows that for each [, π), x R and < r < 1, the image under p W of the corresponding core curve γ x,r is the graph of a linear or quadratic function f x,r over an interval I x,r. Lemma 4.5. For all [, π), p = (x, ) with x R, and < r < 1, we have p W (B H (p, r)) N E (p W (γ x,r ), 5r ), where the expression on the right denotes the Euclidean 5r -neighborhood of p W (γ x,r ). More precisely, (4.5) p W (B H (p, r)) {(α, τ) R : α I x,r, f x,r (α) 5r τ f x,r (α) + 5r }. Proof of Lemma 4.5. We discuss the proof for the case (, π ]. The other cases can be 4 treated similarly, using the appropriate core curve. For an arbitrary point (x + iy, t ) in the ball B H (p, r), one finds (4.6) x x r, y r and t + x y r. The projection is given by p W (x + iy, t ) = ( x sin + y cos, t + (x y ) sin x y cos ) =: (α, τ ). For points on the core curve, (x + iy, x y) γ x,r, we have p W (x + iy, x y) = ( x sin + y cos, x y + (x y ) sin x y cos ) =: (α, τ). Thus, as a subset of R, the set p W (γ x,r ) coincides with the graph of the function (4.7) f x,r (α) = tan (α + x sin ) + x ( tan ) sin cos over the interval I x,r = [ x sin r cos, x sin + r cos ]. 1

13 The goal is now to find a point in p W (γ x,r ) which lies close to p W (x + iy, t ). To this end, let (4.8) y := y (x x ) tan and note that y y + x x tan (1 + tan )r r. It follows (x + iy, x y) γ x,r. We claim that the Euclidean distance between the points p W (x + iy, x y) and p W (x + iy, t ) is at most 5r. First, we observe α α = (x x ) sin + (y y ) cos = (x x ) sin (x x ) tan cos = for y as in (4.8). Second, we compute τ τ = x y t + (x y x + y ) sin cos (x y x y ). Inserting y from (4.8) and using trigonometric relations yields τ τ = x y t + x (x x ) tan + sin cos (x + y (x x ) tan (x x ) tan x ) (cos sin )(x y x (x x ) tan x y ) = (t + x y ) + x (x x ) sin cos + x (x x ) sin cos y (x x ) sin3 cos (x x ) = (t + x y ) (x x ) sin cos y (x x ) sin3 cos (x x ) = (t + x y ) (x x ) tan y (x x ). Hence, from (4.6) it follows, τ τ t + x y + tan x x + y x x (1 + tan + )r. For (, π] this yields (α α 4 ) + (τ τ ) 5r which concludes the proof in this case. The proof for [ 3π, π) is very similar. We employ again the formula (4.7). For 4 = we have to consider a linear function instead of the quadratic function (4.7). The case ( π, 3π ) can be treated similarly, starting from a core curve of the second type. 4 4 Lemma 4.6. Let [, π) and R >. There exist constants c 1 > and c = c (R) > such that for all < r < 1, z = z e i C with z R and t R, the set p W (B H ((z, t ), r)) can be covered by M balls B W (p j, c 1 r ) := B H (p j, c 1 r ) W, j {1,..., M}, with M c r 3. Proof. Since the restriction of the Heisenberg metric to the vertical plane W is comparable to the parabolic heat metric on R, there exists a constant c 1 > such that R(p, r ) := {(α, τ) R : α α r, τ τ r 4 } B W (p, c 1 r ) for all p = (α, τ ) W and r. It is therefore enough to construct a cover by rectangles R(p j, r ), j {1,..., M}. Moreover, it suffices to prove the result for balls centered on the x-axis, i.e., for balls B H ((z, t ), r) with z = x R and t =. Indeed, an arbitrary ball B H ((z, t ), r) can be 13

14 obtained from B H (( z, ), r) by a (Euclidean) vertical translation to height t and a rotation about the t-axis with rotation angle. Then, as a subset of R, the image p W (B H ((z, t ), r)) coincides with a vertical translation of p W (B H (( z, ), r)). Let us consider a ball with radius r < 1, centered at a point p = (x, ) with x R, x < R. The goal is to cover the set S (x, r) := {(α, τ) R : α I x,r, f x,r (α) 5r τ f x,r (α) + 5r } W which, by Lemma 4.5, contains the image of B H (p, r) under p W, efficiently by rectangles R(p j, r ), j {1,..., M}. Let us assume that f x,r is defined on the entire real line. For given and x, we fix a particular point { x α :=, if (, π] [ 3π, π), sin 4 4, else. In the case where f x,r is a quadratic function, it has an extremal point at α. This is shown in the proof of Lemma 4.5 for the case (, π] [ 3π, π). We write 4 4 (I x,r ) k := [α + kr, α + (k + 1)r ). It can be checked that the interval I x,r has length at most 4r (this is done explicitly in the proof of Lemma 4.5 for the case (, π]), whereas each interval of the form 4 (Ix,r ) k has length r. It follows that I x,r has nonempty intersection with at most (4.9) N 6 r of the disjoint intervals (I x,r ) k. Let k be such that I x,r (I x,r ) k. Consider now the portion of S (x, r) which lies above the interval (I x,r ) k, more precisely, {(α, τ) S (x, r) : α (I x,r ) k }. In order to see how many rectangles R(p j, r ) we need to cover this set, we have to estimate its vertical height. direct computation for the several possible cases shows that there exists a constant c = c (R) such that for each k Z with I x,r (I x,r ) k there is an interval ) k of length c r with (J x,r (4.1) {(α, τ) S (x, r) : α (I x,r ) k } (I x,r ) k (J x,r ) k. Hence, because of (4.5) and (4.1), there exists an integer N N and points in the vertical plane W with p k,l = (α k,l, τ k,l ), l {1,..., N } (4.11) N c + 1 r such that p W (B H ((x, ), r)) ((I x,r ) k R) N R(p k,l, r ). From (4.9) and (4.11) it follows that the image p W (B H ((x, ), r)) can be covered by M := N N 6(c + 1) r 3 14 l=1

15 rectangles R(p j, r ) = R(p k,l, r ). Since each of these rectangles is contained in a ball B W (p j, c 1 r ), this concludes the proof of Lemma 4.6. Proof of Proposition 4.3. We may without loss of generality assume that the set is bounded. We denote its Hausdorff dimension by dim H = s [1, 3]. Then we have H s+ɛ H () = for all ɛ > and thus, for each ɛ >, with (H H ) s+ɛ δ () = for all δ >. Hence, for all ɛ > and < δ < 1, there exists a countable collection of balls B H (p i, r i ), i N, r i δ (4.1) i N B H (p i, r i ), i=1 r s+ɛ i < δ. We write p i = (z i, t i ) = ( z i e i,i, t i ). Since the set is bounded, we may assume that there exists R > such that z i R for all i N. Fix now [, π). It follows from Lemma 4.6 that there exist c 1, c > (independent of p i and r i ), constants M i with M i c, and points p ri 3 i,j, i N and j {1,..., M i } such that (4.13) p W () i N For σ, notice that M i j=1 B W (p i,j, c 1 r i ). diam(b W (p i,j, c 1 ri )) σ+ɛ = M i 1 ri i,j i N j=1(c ) σ+ɛ = M i (c 1 ) σ+ɛ r σ+ɛ i i N (c 1 ) σ+ɛ c r (σ 3)+ɛ i. Now if σ is chosen such that σ 3 = s, i.e., i N σ = 1 (s + 3) = 1 (dim H + 3), it follows (4.14) From (4.13) and (4.14), we conclude that diam(b W (p i,j, c 1 ri )) σ+ɛ < (c 1 ) σ+ɛ c δ. i,j Letting δ tend to zero yields and thus, (H H ) σ+ɛ c 1 δ (p W ()) (c 1 ) σ+ɛ c δ. H σ+ɛ H (p W ()) = dim H p W () σ = 1 (dim H + 3), as desired. This concludes the proof of Proposition

16 Next, we discuss universal lower dimension bounds for vertical projections. We will prove two propositions. Proposition 4.7 is the vertical analog of Lemma 3.1. Observe that the failure of the vertical projection to be Lipschitz resurfaces in the proof of this result; see (4.15). Proposition 4.9 uses a slicing theorem for dimensions of intersections of sets with planes in Euclidean space together with the dimension comparison principle. Taken together, these two propositions establish the universal lower bounds in Theorem 1.3. Proposition 4.7. Let H be Borel with dim H 1. Then dim H p W () 1 (dim H 1) for every. In the proof, we use the following elementary estimate whose proof we omit. Compare Lemma 4.4 in [9]. Lemma 4.8. There exists an absolute constant C > so that a 1 b a 4 H b 4 H + C b H whenever a and b are points in H with a H 1 and b H 1. Proof of Proposition 4.7. We may assume without loss of generality that is bounded. In fact, let us assume that z 1 for all points p = (z, t). Fix [, π), let s > dim H p W (), let ɛ >, and cover the set p W () with a family of Heisenberg balls {B H ((z i, t i ), r i )} i so that i rs i < ɛ. Claim: We can choose C > and N i C r 1/ i values (z ij, t ij ) contained in the fiber p 1 W (z i, t i ) so that the family {B H ((z ij, t ij ), C ri )} j covers p 1 W (B H ((z i, t i ), r i )). To prove the claim, it suffices to prove that (4.15) p 1 W (B H (q, r) W ) B H ((, ), 1) N H (q V, C r) for some constant C >, whenever q W and < r 1. Here N H (S, δ) denotes the δ-neighborhood of a set S H in the metric d H, i.e., N H (S, δ) = s S B H(s, δ). The inclusion in (4.15) is a consequence of the following statement: For all q W so that d H (q, q ) r and for all p V so that p H 1, there exists p V so that d H (q p, q p) C r. To establish this statement, choose p = p. Then d H (q p, q p) 4 = p 1 (q 1 q ) p 4 H d H (q, q ) 4 + Cd H (q, q ) by Lemma 4.8. Since d H (q, q ) r by assumption and r 1, we conclude that which finishes the proof of the claim. d H (q p, q p) 4 Cr With the claim in hand, the rest of the proof of Proposition 4.7 proceeds exactly as for its horizontal counterpart. The family {B H ((z ij, t ij ), C ri )} i,j covers the set and we compute rad(b H ((z ij, t ij ), C ri )) s+1 = N i (C ri ) s+1 C s+ ri s C s+ ɛ. i i i,j Letting ɛ gives H s+1 H () = so dim H s + 1. Letting s dim H p W () completes the proof. 16

17 Proposition 4.9. Let H be Borel with dim H 3. Then dim H p W () dim H 5 for every. Proof. It suffices to assume that dim H > 3. By the dimension comparison principle, dim E dim H 1 >. Let < ε < dim H 3. ccording to the classical Euclidean intersection theorem (see Theorem 1.1 in [17]), there exists a plane Π in R 3 for which dim E ( Π) dim E 1 ε dim H ε > 1. Furthermore, we may assume that Π is not a vertical plane, i.e., Π is a t-graph: the graph of a function u : C R. Let us write Π = {(z, t) : t = u(z) := Re(az) + b} for some a C and b R. Consider the map F : R R given as the composition of the graph map id u, the vertical projection p W, and the coordinate chart ϕ W (see (1.3)). Written in complex notation, The Jacobian determinant of F is given by F (z) = (Im(e i z), Re(az) + b Im(e i z )). det DF = Im(e i (z ia)) and the restriction of p W to {(z, u(z)) Π : det DF (z) } is locally bi-lipschitz. Observe that γ = {(z, u(z)) Π : det DF (z) = } is a line. Since dim E ( Π) > 1, dim E ( (Π \ γ)) = dim E ( Π) and it follows that (4.16) dim E p W () dim E p W ( (Π \ γ)) = dim E ( (Π \ γ)) = dim E ( Π) dim H ε. To complete the proof, we use the dimension comparison principle again to switch back from the Euclidean dimension of the projected set to its Heisenberg dimension. Since p W () is contained in W, we can use the improved lower dimension comparison bound from Theorem.8. Using (4.16) we obtain and thus, letting ε tend to zero, dim H p W () β W (dim E p W ()) β W (dim H ε), dim H p W () dim H 5, as asserted in the statement. The proof is complete. The result of Theorem 1.3 follows by combining Proposition 4., 4.7 and 4.9. We now turn to the proof of the sharpness statement of Theorem 1.3. Proposition 4.1. In each of the following statements, the set is a compact subset of H. (a) For all β 1 there exists so that dim H = β and dim H p W () =. (b) For all 1 β 3 there exists so that dim H = β and dim H p W () = (β 1)/. (c) For all 3 β 4 there exists so that dim H = β and dim H p W () = β 5. (d) For all β 1 there exists so that dim H = β and dim H p W () = β. (e) For all 1 β 3 there exists so that dim H = β and dim H p W () = 1 (β + 3). (f) For all 3 β 4 there exists so that dim H = β and dim H p W () = 3. 17

18 Proof of Proposition 4.1. For a proof of the statements (d), (e) and (f), i.e. for the sharpness of the upper dimension bounds, see the proof of Proposition 5.3, where sets are constructed for which the corresponding dimension values hold for all directions, and not merely for =. In the following, we discuss the sharpness of the lower dimension bound, i.e., the cases (a), (b) and (c). ssume that β 1. Let V be a compact β-set. Then p W () = {(, )} is zero-dimensional. This gives an example of a set satisfying (a). Examples for (b) and (c) are based on the following special case (β = 3), which we describe first. Let B = {(iy, ) : y R} be the y axis and let (4.17) = p 1 W (B ) = {(x + iy, xy) : x, y R}. Then dim H B = 1, while dim H = 3. Next, assume that 1 < β < 3; we construct a set satisfying (b). The desired set is constructed as a subset of the set defined in (4.17). Let S R be a compact hlfors regular set of dimension (β 1)/ and consider the set = p 1 W ({(iy, ) : y S}). 1 Clearly dim H p W () = dim E p W () = (β 1)/. By Theorem 1.3, (dim H 1) dim H p W () so it suffices to verify that dim H β. gain we will appeal to Theorem.4; the details are similar to those in the proof of Proposition 3.(d). Define a set function µ on as follows: µ(e) = S H 1 H(E L y ) dh (β 1)/ E (y), for Borel sets E, where L y := p 1 W (iy, ). Let B H (p, r) be a ball in (H, d H ) centered at p with radius r. Write p = (x + iy, x y ) for some y S and x R. Lemma If B H (p, r) L y, then y y r and H 1 H (B H(p, r) L y ) Cr, for a constant C independent of p, r and y. ssuming the lemma we complete the proof in this case: µ(b H (p, r)) Cr H (β 1)/ E ({y : B H (p, r) L y }) Cr H (β 1)/ E ([y r, y +r ] ) C r β for C > independent of p and r. Hence µ satisfies the upper mass bound (.) on with exponent β. By Theorem.4, dim H β. It remains to prove the lemma. Suppose that B H (p, r) L y. Then B H (p, r) B H (q, r) for some q B H (p, r) L y and H 1 H (B H(p, r) L y ) H 1 H (B H(q, r) L y ) Cr since L y is a horizontal line. Furthermore, if q = (x + iy, xy) then Since d H (p, q) r we conclude p 1 q = ((x x ) + i(y y ), (x + x )(y y )). x x r and (x + x )(y y ) r. We may restrict to the subset of consisting of points (x + iy, t) for which x 1 and consider only radii r < 1. Then x + x x x x r > 1 and so y y Cr. This completes the proof of Lemma 4.11 and ends the construction when β [1, 3]. 18

19 Finally, assume that 3 < β 4. The desired set in this case is constructed as a union of a collection of vertical translates of the set defined in (4.17). Let S R be a compact hlfors regular set of dimension β 3 and let = s S τ (,s) ( ), where τ q : H H, τ q (p) = q p, denotes left translation by q H. Then H is compact and p W () {(iy, s) : y [, 1], s S}, whence dim H p W () 1 + (β 3) = β 5. By Theorem 1.3, dim H 5 dim H p W (), so it suffices to verify that dim H β. Define a set function µ on by setting µ(e) = HH(E 3 Σ s ) dh β 3 E (s), for Borel sets E, S where Σ s = τ (,s) ( ). Let B H (p, r) be a ball in (H, d H ) centered at p with radius r. Write p = (x + iy, x y + s ) for some s S and x, y R. Then p Σ s. Lemma 4.1. If B H (p, r) Σ s, then s s Cr and H 3 H (B H(p, r) Σ s ) Cr 3. ssuming the lemma we complete the proof in this case: µ(b H (p, r)) Cr 3 H β 3 E ({s : B H(p, r) Σ s }) Cr 3 H β 3 E ([s r, s + r] ) Cr β. Hence µ satisfies the upper mass bound (.) on with exponent β. By Theorem.4, dim H β. It remains to prove the lemma. Suppose that B H (p, r) Σ s. Then B H (p, r) B H (q, r) for some q B H (p, r) Σ s and HH 3 (B H(p, r) Σ s ) HH 3 (B H(q, r) Σ s ) Cr 3 since Σ s is a smooth submanifold. Furthermore, if q = (x + iy, xy + s) then Since d H (p, q) r we conclude p 1 q = ((x x ) + i(y y ), (s s ) + (x + x )(y y )). y y r and (s s ) + (x + x )(y y ) r and so s s r + C y y Cr. This completes the proof of Lemma 4.1 and ends the construction when β [3, 4]. 5. lmost sure bounds for vertical projections The goal of this section is to prove an almost sure lower bound for vertical projections (Theorem 1.4) and to verify that the given universal upper bound is sharp even as an almost sure statement. The arguments concerning the lower bound go along the lines of the proof of the corresponding Euclidean result. However, it is considerably more difficult to establish the integrability of certain functions given in terms of the Heisenberg distance between projected points and the proof works only for a restricted range of dimensions, namely, dim H 1. Here is the main proposition of this section. Proposition 5.1. Let H be a Borel set with dim H 1. Then dim H p W () dim H for a.e.. 19

20 Corollary 5.. Let H be a Borel set. Then dim H p W () min{dim H, 1} for a.e.. To prove the corollary, let be a Borel subset of H with dim H > 1 and choose a subset B with dim H B = 1. For a.e., we have dim H p W () dim H p W (B) dim H B = 1. Proof of Proposition 5.1. Fix < σ < dim H. By Theorem.6, there exists µ M() with I σ (µ) = d H (p, q) σ dµ(p) dµ(q) <. Using this measure, we will define a family of measures {µ } [,π) so that µ M(p W ()) and (5.1) π I σ (µ ) d <. Once this done, the proof is finished by another appeal to Theorem.6 (since the integrand of (5.1) must be finite for almost every ) and by taking the limit as σ increases to dim H. It remains to construct the measures µ and verify (5.1). Consider the pushforward measure µ := (p W ) µ defined by (p W ) µ(e) = µ(p 1 W (E)). It is not hard to see that µ is in M(p W ()). By Fubini s theorem and the definition of the pushforward measure, the integral in (5.1) is equal to π (5.) d H (p W (p), p W (q)) σ d dµ(p) dµ(q). We claim that the quantity in (5.) is bounded above by an absolute constant multiple of I σ (µ), i.e., π (5.3) d H (p W (p), p W (q)) σ d dµ(p) dµ(q) C d H (p, q) σ dµ(p) dµ(q). Unlike the Euclidean case, the distance d H (p W (p), p W (q)) is not related to the distance d H (p, q) in any simple way. This means that we are not able to prove (5.3) by bounding the inner integral pointwise by d H (p, q) σ as in the Euclidean case. The main technical difficulties in the proof lie in the verification of (5.3). In order to prove (5.3), we split the domain of integration into two pieces, according to the two terms which appear in the formula (.8) for the Heisenberg distance. Let and 1 := {(p, q) : z ζ t τ + z ζ sin(ϕ 1 ϕ ) } := {(p, q) : z ζ < t τ + z ζ sin(ϕ 1 ϕ ) }. First, suppose that (p, q) 1. We observe the following distance estimates in this case: Then d H (p, q) 4 z ζ 4 and d H (p W (p), p W (q)) 4 z ζ 4 sin 4 (ϕ 1 ). π d H (p W (p), p W (q)) σ d z ζ σ π d sin(ϕ 1 ) σ C 1d H (p, q) σ, where C 1 = σ/4 π sin σ d <. Note that in this case we use the assumption σ < 1, and also that the constant C 1 is independent of p and q.

21 Next, suppose that (p, q). Let us introduce the abbreviating notation a := z ζ, b := t τ, and ϕ := ϕ ϕ 1. Observe that the condition (p, q) implies that either b is nonzero, or that both a and sin ϕ are nonzero. We also have d H (p, q) 4 (b + a sin ϕ ) and d H (p W (p), p W (q)) 4 (b a sin(ϕ + ϕ 1 )). Hence π and d H (p W (p), p W (q)) σ d π b a sin(ϕ +ϕ 1 ) σ/ d = 1 d H (p, q) σ σ/4 b + a sin ϕ σ/, so it suffices to find a constant C independent of a, b and ϕ for which (5.4) π d b + a sin σ/ C b + a sin ϕ σ/ π b+a sin σ/ d whenever either b or a sin ϕ. We finish the proof by verifying (5.4) for some explicit constant C. If a =, then (5.4) is satisfied for C = π, so assume a. Then (5.4) is equivalent to (5.5) π d r + sin σ/ C r + sin ϕ σ/ where r = b/a. By a change of variables, we may assume without loss of generality that r. Observe that (5.5) is implied by (5.6) π so we are reduced to verifying (5.6). Consider the function π d r + sin σ/ C (1 + r) σ/, g(r) := (1 + r) σ/ π d r + sin σ/. straightforward computation shows that g is monotone decreasing on the interval [1, ), with g(1) = σ/ π (1 + sin ) σ/ d which is finite since σ < 1. Suppose that r < 1 and write r = sin ψ for an appropriate ψ. To obtain a bound which is uniform in ψ, we use a trigonometric identity, the Cauchy Schwarz inequality and change of variables to obtain π ( ) σ/ ( ) σ/ 1 + sin ψ π 1 + sin ψ g(sin ψ) = d = sin ψ + sin sin( ψ+ ψ ) cos( ) d ( π sin( ψ+ ) σ/ cos( ψ ) σ/ d ) 1/ ( π 1/ π sin( ψ+ ) σ d cos( d) ψ ) σ = sin σ d. The latter integral is finite since σ < 1. This completes the proof. 1

22 The lower bound in Theorem 1.4 follows by combining Proposition 4.7, Proposition 4.9 and Corollary 5.. The almost sure upper bound for the vertical projections is the same as the universal upper bound which was proved in Proposition 4., and this bound is sharp. Proposition 5.3. In each of the following statements, the set is a compact subset of H. (a) For all β 1 there exists such that dim H = β and dim H p W () = β for all. (b) For all 1 < β < 3 there exists such that dim H = β and dim H p W () = 1 (β + 3) for all. (c) For all 3 β 4 there exists such that dim H = β and dim H p W () = 3 for all. Proof. First, we construct a set satisfying (a) for every. Let V be a compact β-set. Then p W () has Heisenberg Hausdorff dimension β for every. Compare Example 4.1. To construct an example which works for every value of, let be the union of two compact β-sets, one contained in V and one contained in V π/. It is not hard to find a set with dim H = dim H p W = 3, for every [, π). Take for instance = {(z, ) : z 1}. The case (c) becomes then quite simple. Indeed, given β [3, 4], let C H be any compact β-set which contains the set. Then p W (C) p W () has dimension 3 for every. It remains to discuss the case (b). For a fixed number β (1, 3), we choose a Cantor set C of Euclidean dimension β 1 on the interval [, π) and set := {(re iϕ, ) : r [ 1, 1], ϕ C}. We will prove that dim H = β. The set is made up of horizontal curves, more precisely, radial segments inside the plane t =. For almost every direction, the projection onto a vertical plane will be a non-horizontal parabola. This leads to the desired increase in dimension. To define the set C, we employ the similarity maps S 1 (x) = λx and S (x) = λx + 1 λ on R with λ := β (, 1 ). The resulting invariant set C(λ) = S 1 (C(λ)) S (C(λ)) is a compact subset of [, 1] with < H (β 1)/ E (C(λ)) < and thus dim E C(λ) = log. We set = β 1 log λ C := 8 i=1f i (C(λ)), where f i (ϕ) = πϕ + (i 1) π, and denote further 8 4 i := {(re iϕ, ) : r [ 1, 1], ϕ f i(c(λ))} for i {1,..., 8}. Each set i consists of radial segments of length 1, emanating from a Cantor set on the unit circle. The statement given in (b) then follows from the two subsequent lemmas. Lemma 5.4. The set has dimension dim H () = β. Lemma 5.5. For an arbitrary [, π), the set p W () has dimension dim H (p W ()) = β+3.