Intersections of Projections and Slicing Theorems for the Isotropic Grassmannian and the Heisenberg group

Abstract: This paper studies the Hausdor dimension of the intersection of isotropic projections of subsets of R2n, as well as dimension of intersections of sets with isotropic planes. It is shown that if A and B are Borel subsets of R2n of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of A and B under orthogonal projections onto these planes have positive Hausdor m-measure. In addition, if A is a measurable set of Hausdor dimension greater than m, then there is a set B Ă R2n with dim B ď m such that for all x P R2nzB there is a positive measure set of isotropic m-planes for which the translate by x of the orthogonal complement of each such plane, intersects A on a set of dimension dimA ́ m. These results are then applied to obtain analogous results on the nth Heisenberg group.


Introduction
For m ď n, denote by Gpn, mq the Grassmannian of m-dimensional subspaces of R n . Gpn, mq is endowed with a unique Opnq-invariant probability measure γn,m. These spaces play an important role in the integral geometry of Euclidean space. A lot of work has been done concerning the e ect of orthogonal projection maps, P V : R n → V, on the dimension and measure of sets. One of the most in uential results dates back to J. Marstrand [12] who proved that for a Borel set A Ă R , dim P V A " mintdim A, u for γ , -a.e. V P Gp , q. This was later generalized to higher dimensions by P. Mattila in [13] and later work. The most general result, including the Besicovitch-Federer characterization of unrecti ability ( [3], [8]), can be stated in the following encompassing theorem. Moreover, in the case where s " m, and with the added hypotesis that H m pAq ă 8 , H m pP V Aq " for γn,m-a.e. V P Gpn, mq if, and only if, A is purely m-unrecti able.
1. If s ď m, dim P V A " s for µ n,m -a.e. V P G h p n, mq. 2. If s ą m, H m pP V Aq ą for µ n,m -a.e. V P G h p n, mq. 3. If s ą m then, IntpP V pAq ‰ ∅, for µ n,m´a .e.V P G h p n, mq.
In the case when s " m, H m pP V Aq " for µ n,m -a.e. V P G h p n, mq if, and only if, A is purely m-unrecti able.
Parts (1) and (2) were originally proven in [2]. The later proof using transversality also provides dimension estimates for the subset of isotropic planes where the theorem fails.
In this work, the results will be analogous statements to Theorems 1.2 and 1.3. The following are the main results. Theorem 1.5. Let A, B Ă R n be Borel sets, and for V P G h p n, mq let P V be the orthogonal projection onto V. 1. If dim A ą m and dim B ą m, then µ n,m pV P G h p n, mq : H m pP V A X P V Bq ą q ą .
2. If dim A ą m and dim B ą m, then µ n,m pV P G h p n, mq : IntpP V A X P V Bq ‰ ∅q ą .
3. If dim A ą m, dim B ď m, and dim A`dim B ą m, then for all ϵ ą , µ n,m pV P G h p n, mq : dimrP V A X P V Bs ą dim B´ϵq ą .
And, Theorem 1.6. Let A Ă R n be a Borel set such that for some m ă s ď n, ă H s pAq ă 8. Then, there is a Borel set B Ă R n with dim B ď m, such that for all x P R n zB, µ n,m pV P G h p n, mq : dimrA X pV K`x qs " s´mq ą .
Since the tP V : V P G h p n, mqu forms a transversal family, it is not unexpected that results concerning the family of all orthogonal projections tP V : V P Gpn, mqu, carry over to the isotropic case. This work is a con rmation of this intuition, and many of the proof are very simple adaptations of the arguments used in [18]. These results, however, let us obtain similar results in the context of the Heisenberg group H n . As noted by P. Mattila, R. Serapioni and F. Serra-Cassano in [19] and later used by Z. Balogh, K. Fässler, P. Mattila, and J.T. Tyson in [2], the manifolds G h p n, mq play a crucial role in the horizontal geometry of the Heisenberg group. It is therefore not surprising that the results obtained for G h p n, mq in this paper have direct analogues in the Heisenberg group. For instance, Theorem 1.7. Let A, B Ă H n be Borel sets, let V denote the horizontal subgroup corresponding to the isotropic plane V P G h p n, mq, and let P V denote the homogeneous projection onto V.
1. If dim A, dim B ą m` then, µ n,m pV P G h p n, mq : H m pP V A X P V Bq ą q ą .
2. If dim A, dim B ą m` then, µ n,m pV P G h p n, mq : IntpP V A X P V Bq ‰ ∅q ą .
3. If dim A ą m` , dim B ď m` but dim A`dim B ą m` , then µ n,m pV P G h p n, mq : dimrP V A X P V Bs ą dim B´ ´ϵq ą .
In [2], the authors were able to obtain a slicing theorem for H n along the lines of Theorem 1.3. In their paper, they consider slices by right cosets of vertical planes and obtained, Theorem 1.8. Let A Ă H n be a Borel set with dim A " s ą m` and such that ă H s pAq ă 8. Then H m pu P V : dimrA X pV K˚u qs " s´mq ą , for µ n,m´a .e.V P G h p n, mq.
Using the results for G h p n, mq obtained in this paper, it is possible to prove very similar result to this last theorem and further obtain a dimension bound for the set of exceptions to this slicing result. Theorem 1.9. Let s P R be such that m` ă s ď n` , and A Ă H n be a Borel set with ă H s pAq ă 8. Then, there is a Borel set B Ă H n with dim B ď m` , such that for all p P H n zB, µ n,m pV P G h p n, mq : dimrA X pV K˚p qs " s´mq ą . (1.1)

Preliminaries . Hausdor measure and dimension
Given a metric space pX, dq and a subset A Ă X we will denote by H s pAq the Hausdor s-measure of A. That is, The Hausdor dimension of A is de ned as dim A " infts ą : H s pAq " u " supts ą : H s pAq " 8u. Aside from the de nition, there are other computational tools to estimate Hausdor dimension of sets. One of the most useful ones is Frostman's lemma, proven by Otto Frostman as part of his dissertation. A constructive proof of Frostman's lemma for compact sets is given in [15,Theorem 8.8]. The more general case of Suslin sets was originally treated by Carleson in [6]. Frostman's lemma can also be used to obtain another powerful tool which allows the computation of Hausdor dimension of a set in terms of energies of measures supported on that set. This is sometimes referred to as the "energy version" of Frostman's lemma. This approach was rst introduced in R by R. Kaufman in [11]. A simple proof of this is given in [16,Theorem 2.8].

. Fourier transform of measures
In the same way as for L functions, one can de ne the Fourier transform of a nite measure as p µpξ q " ş R n e´ πix¨ξ dµ. Note that with this de nition, p µ is a bounded Lipschitz function. The convolution of a function and a measure is de ned as whenever the integral is de ned. More generally, convolution of nite measures is de ned as another nite measure, µ˚ν, given by for all non-negative, continuous bounded functions φ (see [15, pp.16]). The following properties hold,

If p
µ P L pR n q, then µ " fdx for some f P L pR n q. 5. If p µ P L pR n q, then µ " gdx for some g P CpR n q.
One of the main connection between Fourier analysis and Hausdor dimension comes from using the Fourier transform to estimate energy integrals. For ă s ă n, we denote by Rs the kernel of the Riesz potential, Rspxq " |x|´s. The energy of a measure µ can be written as Ispµq " ş Rs˚µdµ. Using the distributional Fourier transform and its properties, one can compute p Rs " κpn, sqR n´s , where κ is an universal constant depending on s and n. Using these results together, smooth approximations to the measure µ and the properties of the Fourier transform discussed above, one can justify the following formula, A proof of this can be found on [16, pp. 38]. Finally, we will also make use of the mutual energy of measures. For compactly supported measures µ and ν we de ne the mutual s-energy, with ď s ď n, as Just as with energies of measures, it is convenient to write the mutual energy as an integral on the frequency domain. If Iαpµq ă 8, I β pνq ă 8 and s " α`β , by Holder's inequality we have, Therefore, if tψ δ u δą Ă SpR n q is an approximation to the identity, and we take µ δ " ψ δ˚µ , ν δ " ψ δ˚ν , the fact that p ψ δ is uniformly bounded tells us that |p µ δ p ν δ ||¨| s´n " |p µp ν|| p ψ δ | |¨| s´n , is dominated by |p µp ν||¨| s´n . This validates the formula Ispµ, νq " κpn, sq ż p µpxq s p νpxq|x| s´n dx, (2.5) for s ď α`β . Also note that since µ and ν are positive measures the mutual energy is positive.

Orthogonal projections onto isotropic planes
We begin this section by establishing already known results about dimension distortion by isotropic projections. The most natural statement is that analogous to Theorem 1.1 and we state it in its strongest form as follows. This was proven by proven by R. Hovila in [10], and implies the almost everywhere statements of Theorem 1.4.
With these general result about projections onto isotropic planes in hand, we now turn our attention to intersections of such projections. We would like to establish a result analogous to Theorem 1.2. In order to establish such a result we will need an isotropic version of the Grassmannian disintegration fromula in R n . This is precisely the content of the following Lemma 3.2 (Isotropic disintegration formula). There exists a positive constant c " cpn, mq such that, for all f P L pR n q, Proof. Using spherical coordinates on V we can compute the integral on the left as follows, Now we take a closer look at the inner double-integral. We know that Upnq acts transitively on S n´ , therefore, up to multiplication by a constant, there is a unique Upnq-invariant measure on S n´ . Since σ n´ is Opnq invariant, it is in particular Upnq invariant. Hence any Upnq-invariant measure on S n´ must be a constant multiple of the surface measure σ n´ . Now, for a function φ on S n´ , one can check that the measure on S n´ given by, Going back to our overall integral, we now have, One application of this formula is the following lemma, which will be useful later, Lemma 3.3. Let m ă s ď n and µ P MpR n q be a measure such that, for some C ą , µpBpx, rqq ď Cr s for all x P R n , r ą .
Then, P V# µ ! H m for µ n,m´a lmost every V P G h p n, mq.
Proof. Using [15,Theorem 1.15], for any σ ă s we have In particular Impµq ă 8. By Lemma 3.2, and using the fact that for any plane V and v P V, { P V# µpvq " p µpvq, we compute That is, { P V# µ P L pVq for µ n,m´a lmost every V P G h p n, mq. It follows from part 4 on Lemma 2.3 (see also [16,Theorem 3.3]), that P V# µpvq ! H m with density in L pVq, for µ n,m´a lmost every V P G h p n, mq.
Part (1) of Theorem 1.5 is proven next.
Proof. Pick µ P MpAq and ν P MpBq both with nite m energy. As seen in the proof of Lemma 3.3 we know that for µ n,m -a.e. V, the measures P V# µ and P V# ν are L pVq functions, we denote these functions by µ V , and ν V . By Hölder's inequality, the product µ V ν V is in L pVq for µ n,m -a.e. V. Now we note that if To this end, we note that for a P V, p µ V paq " p µpaq, and similarly for ν. Applying Plancherel's theorem on V, and Lemma 3.2 we get, To prove part (2), a statement at the level of measures is proven rst. This will directly imply the desired result.
Theorem 3.5. Assume µ, ν are compactly supported Radon measures in R n with nite s and t energy respectively.
1. If s`t " m, µ n,m pV P G h p n, mq : 2. If s ą m, and t ą m, then µ n,m pV P G h p n, mq : It follows from Lemma 3.2, together with the fact that, for v P V , P V# µpvq " p µpvq and { P v# νpvq " p νpvq, that { P V# µ z P V# νdH m ą for a µ n,m´p ositive measure set of planes V P G h p n, mq. Since sptpg V q " sptp Č P V# µq`sptpP V# νq " sptpP V# νq´sptpP V# µq, and P sptpg V q, it follows that there is u P sptpP V# µq and v P sptpP V# νq such that " v´u, or equivalently, v " u. Hence, sptpµ V q X sptpP V# νq ‰ ∅. This proves the claim.
2. Now assume that s, t ą m. Then, by Theorem 3.4 in [16], both P V# µ and P V# ν have continuous densities µ V and ν V for µ n,m -a.e. V. Since, by (3.1) we know that for a µ n,m -positive set ş V µ V ν V dH m ą , it follows that for such µ n,m -positive set, µ V ν V remains positive in some open subset of V. Since sptpµ V ν V q Ă sptpµ V q X sptpν V q, the theorem follows.
As mentioned above, as a direct corollary of part (2) of Lemma 3.5, we obtain part (2) of Theorem 1.5. Now lets proceed with the proof of part (3) of Theorem 1.5.
Proof. We begin as usual, by choosing s, t P R such that m ă s ă dim A, ă t ă dim B and s`t ą m. By Frostman's lemma, one can then pick µ P MpAq and ν P MpBq with nite s and t energies respectively. Since s ą m we have that for µ n,m´a .e. V , P V# µ ! dH m with density µ V P L pVq. Following the same lines as the proof of part (1) of Theorem 3.5, we aim to nds a family of measures ρ V P MpP V pAq X P V pBqq, but this time we also require that I t pρ V q ă 8. In principle, we would like to use the family of measures ρ V " µ V dP V# ν. However, a priori, we do not know that µ V P L pP V# νq. Instead, let µ δ " µ˚ψ δ be the standard convolution approximation to µ, where ψ is smooth and compactly supported in Bp , q. Using Plancherel's theorem and Lemma 3.2 we have, Hence the right hand side of (3.2) goes to κ pn, mqImpµ, νq. By the choice of µ and ν, and since s`t ą m, we know ă Impµ, νq ă 8. Hence, D c, C ą such that @ δ ą , The aim is now to show that µ V P L pP V# νq and that ĳ µ V dP V# νdµ n,m " lim δ→ ĳ P V# µ δ dP V# νdµ n,m . (3.4) This, together with (3.3) would show that To prove (3.4) we follow a similar argument to the one used in [18] for the analogous statement.
First not that since m´ s´m " m´s ă t, and I m´s pP V# νq À I t pP V# νq, using Lemma 3.2 Next, note that we can write By the dominated convergence theorem it follows that for µ n,m´a lmost every V P G h p n, mq the sequence P V# µ δ converges to f V :" µ V t sptpP V# νq in L pP V# νq. That is to say, the sequence of function With this, consider the measure f V P V# ν. By (3.7) we know that for µ n,m -positively many V , f V is positive and nite on a set of positive P V# ν measure. Therefore f V P V# ν is a non-trivial measure supported on sptpµ V qX sptpν V q Ă P V pAqXP V pBq. That is, f V P V# ν P MpP V pAqXP V pBqq. For each such V, pick a large enough constant C V so that the measure 1 tf V ďC V u f V P V# ν is still non-trivial, and so that 1 tf V ďC V u f V P V# ν has nite t-energy. Since t can be chosen arbitrarily close to dim B the claim follows.

Intersections with isotropic planes
Another problem, which is related to projection theorems, and that has been studied in Euclidean space by many, for instance P. Mattila in [13], Mattila and Orponen in [18] among others, and even in non-Euclidean setting by Balogh, Fässler, Mattila and Tyson in [2], is that of planar slices of sets. We aim to study this problem in Euclidean space but restricted to slices by isotropic planes. The general question can be stated as follows, for A Ă R n with dim A " α what can be said about the "size" (i.e. measure and/or dimension) of the "slice" A X pV K`x q where V P G h p n, mq and x P R n ? Ideally one would hope to obtain a result along the lines of Theorem 1.3. It is important to remark that by V K it is meant the standard orthogonal complement of V P G h p n, mq, in particular, V K need not be isotropic.
As discussed before, for A Ă R n with dim A " α ą m we have that H m pP V Aq ą for µ n,m -a.e. V P G h p n, mq. This implies that the set v P V such that A X P´ V pvq ‰ ∅, has positive m-measure for almost all V. Note that P´ V pvq " V K`v , this hints at the close relation between this question and the questions about projections. This is all encapsulated in Theorem 1.6 which is recalled here, Theorem 1.6. Let A Ă R n be such that for some m ă s ď n, ă H s pAq ă 8. Then there exists a Borel set B Ă R n with dim B ď m, such that for all x P R n zB, µ n,m pV P G h p n, mq : dimrA X pV K`x qs " s´mq ą .
In order to prove this, we will use "sliced" measures. For the sake of completeness the construction of sliced measures context of the isotropic Grassmannian is included. A detailed overview of these measures in the context of the standard Grassmannian is contained in [15, Section 10.1]. Fix V P G h p n, mq. Note that since V is isotropic, m ď n because the maximal dimension of an isotropic plane is half the dimension of the space. Therefore dim V K " n´m ě n, in particular, unless m " n, V K cannot be isotropic. Nevertheless, we endow G h p n, mq K with a "natural" measure, µ K n,m , via µ K n,m pΩq " µ n,m pΩ K q, where Ω K " tV K : V P Ωu. One property that will be of great importance for dimension estimates is the following measure bound, µ n,m ptV : |P V pxq| ď δuq ď cδ m |x|´m (4.1) A proof of this fact can be found in [2]. Given a measure µ P MpR n q and function φ P Cc pR n q one can de ne a new measure by µφpAq " ş A φdµ. We can see that P V# µφ is a measure in MpVq so by the di erentiation theorem, the derivative exists and is nite for H m -a.e. v P V. Here NpV K`v , δq is the δ neighborhood around the plane V K`v . One can check, after some work, that µ V K`v de nes a positive linear functional on Cc pR n q. Therefore, by Riesz representation theorem we can associate a positive Radon measure to µ V K`v that we denote in the same way. As mentioned before, the reader is referred to section 10.1 of [15] for the details. This measure is now supported on pV K`v q X sptµ. Given two Radon measures, λ and γ, the Radon-Nikodym derivative dλ dγ satis es for all Borel sets A, and with equality whenever λ ! γ (Theorem 2.12, [15]). Therefore, for any Borel set B Ă V the measure µ V K`v satis es, for any Borel function g, and with equality whenever P V# µ ! H m . In this case, taking g " 1 P´ V pBq , However, denoting by µ V the density of P V# µ, the de nition of the push-forward measure tells us that Since this is true for any Borel set B Ă V, for H m´a lmost every v P V In particular, if P V# µ ! H m then, Using Lemma 3.2 one may bound the energies I s´m pµ V K`v q by Ispµq as follows, The two inner integrals can be bounded applying (4.2) where B " tv P V : dpx, V K`v q ď δu so that P´ V pBq " ty : |P V px´yq| ď δu. Hence we get, The following lemma, establishes the dimension upper bound for Theorem 1.6.

Lemma 4.2.
Let A Ă R n be a Borel set with ă H s pAq ă 8 for some m ă s ď n. Then, for all x P R n , dimrA X pV K`x qs ď s´m for µ n,m´a .e.V P G h p n, mq.
Proof. The proof begins with the following, Claim: It su ces to show that for every x P R n and r ą , ż˚H s´m rpAzBpx, rqq X pV K`x qsdµ n,m pVq À r´mH s pAzBpx, rqq. Indeed, if 4.7 holds then for each j P N ż˚H s´m rpAzBpx, j qq X pV K`x qs À j´m H s pAq ă 8, so that dimrpAzBpx, j qq X pV K`x qs ď s´m. Since by the countable stability of Hausdor dimension the Lemma follows. All is left is to prove (4.7). This is done using the same argument as in [14, Theorem 6.5]. First, by translating by x, we may assume x " . For each k P N, pick balls B kj such that Since s ą m we get ż G h p n,mq diampB kj X V K q s´m dµ n,m À p r q´mpdiamB kj q s .
By the de nition of H s and Fatou's lemma, This nishes this argument.
This aids the proof of the following result which will prove useful in the proof of the dimension lower bound in Theorem 1.6. Proof. Suppose there is a H s´p ositive measure subset of A such that for every x in that subset dimrA X pV K`x qs ă s´m for a µ n,m´p ositive measure subset of G h p n, mq.
By regularity of H s we may assume such subset is compact. This is to say, there is some m ă σ ă s and compact F Ă A such that H s pFq ą and @ x P F, µ n,m ptV : dimrA X pV K`x qs ă σ´muq ą .
So there exist a compact set G Ă G h p n, mq such that µ n,m pGq ą and for all V P G, µptx : dimrA X pV K`x qs ă σ´muq ą .
By Lemma 3.3, P V# µ ! H m for µ n,m´a lmost every V P G h p n, mq, and since P V# µptv P V : dimrA X pV K`v qs ă σ´muq " µptx P R n : dimrA X pV K`x qs ă σ´muq ą , we get that for V P G H m ptv P V : dimrA X pV K`x qs ă σ´muq ą . Now, if µ V K`v pR n q ą with dimrA X pV K q`vs ă σ´m one has I σ´m pµ V K`v q " 8, so by (4.6) we get which contradicts Lemma 4.1. Hence, it must be that for H s´a lmost every x P A dimrA X pV K`x qs ě s´m for µ n,m´a .e.V P G h p n, mq.
Combining this with Lemma 4.2, we have that for H s´a lmost every x P A dimrA X pV K`x qs " s´m, for µ n,m´a .e. V P G h p n, mq.
By Tonelli's theorem, one may change the order of the measures at will to obtain that the dimension equality holds almost surely in the product measure space AˆG h p n, mq.
In view of (4.7) one could expect that the similar statement: For all x P R n , H s´m rA X pV K`x qs ď 8, for µ n,m´a .e. V P G h p n, mq, holds. I have not been able to prove this, but the following weaker result holds. To see this claim, let λ be the measure on R n given by dλ " 1 NpV ,Rq dH n . In other words λ " H n t NpV ,Rq .
Then there is a constantc "cpR, m, nq, independent of V, such that P V# λ " cH m . Indeed, since NpV , Rq " P´ V K pB V K p , Rqq, for any measurable set S Ă V we have that wherecpR, m, nq is the product of H n´m´m easure of the ball of radius R in R n´m with the constant c such that H n " cH n´mˆHm . Now, by Eilenberg's lemma, Tonelli's theorem, and the fact that for any x P R n , where the last line follows by integrating (4.9) over G h p n, mq with respect to the probability measure µ n,m . This proves the claim.
Now the theorem is proven by contradiction. Suppose there is a set Ω Ă R n such that H n pΩq ą , and for all x P Ω, H s´m pA X pV K`x qq " 8, for a positive µ n,m measure set of planes V P G h p n, mq. By inner regularity of H n we may assume, without loss of generality, that Ω is compact. Then there is R ą such that Ω Ă Bp , R q. Since P V it follows that Bp , R q Ă NpV , R q for every V P G h p n, mq. Therefore, Ω Ă NpV , R q for every V P G h p n, mq. Hence, for every x P Ω H s´m rA X pV K`x qs1 NpV ,R q pxq " H s´m rA X pV K`x qs " 8, for a positive µ n,m measure set of planes v P G h p n, mq. This contradicts the claim and nishes the proof.
The main slicing result is proven next.
Proof of Theorem 1.6. In view of Lemma 4.2, we only need to check that all x outside of a set of dimension less than m, the upper bound dimrA X pV K`x qs ě s´m holds for a µ n,m´p ositive measure set of planes V P G h p n, mq.
Without loss of generality, by Borel regularity of H s we may assume that A is compact. Let B " tx P R n : µ n,m pV P G h p n, mq : dimrA X pV K`x qs ě s´mq " u, that is to say, for all x P B, dimrA X pV K`x qs ă s´m, for µ n,m´a .e. V P G h p n, mq. Since A is compact, the function px, Vq → dimrA X pV K`x qs is a Borel function (see [17]), which shows the set B is a Borel set. Suppose, to obtain a contradiction, that dim B ą m. Pick ν P MpBq with nite m´energy. Let µ " H s t A P MpAq, possibly restricted to a further subset, so that µpBpx, rqq À r s for all x P R n , r ą . In particular, by Lemma 3.3, Impµq ă 8. By assumption, for ν´almost every y P R n dimrA X pV K`y qs ă s´m for µ n,m´a lmost every V P G h p n, mq. Applying Tonelli's theorem to the measures, we get that for µ n,m´-almost every V P G h p n, mq dimrA X pV K`y qs ă s´m for ν´a.e. y P R n . (4.10) Similarly, by Theorem 4.3, for µ n,m´a lmost every V P G h p n, mq we have dimrA X pV K`x qs ě s´m for µ´a.e. x P R n . (4.11) To nd a contradiction one tries to nd y P sptpνq, x P sptpµq satisfying (4.10) and (4.11) respectively and such that V K`x " V K`y , or equivalently, P V pxq " P V pyq. As seen Lemma 3.3, for µ n,m´a lmost every V P G h p n, mq, both P V# µ and P V# ν are absolutely continuous with respect to H m , with densities µ V , ν V P L pVq. For such V de ne A V :" tx P R n : dimrA X pV K`x qs ě s´mu B V :" ty P R n : dimrA X pV K`y qs ă s´mu For µ n,m´a lmost every V, µpR n zA V q " , and νpR n zB V q " . By (3.1), for a µ n,m´p ositive measure set of planes, H m pC V q ą . So we can pick V P G h p n, mq for which all three things are simultaneously satis ed. By (4.6) we have By (4.5) we also have ă µ V pvq " µ V K`v pR n q, and ă ν V pvq " ν V K`v pR n q, so that µ V K`v pA V q ą and ν V K`v pB V q ą . This shows there is v P C V such that µ V K`v , and ν V K`v are both positive. Hence, we can nd x P A V , y P B V , such that P V pxq " P V pyq " v as needed.

Applications to the Heisenberg group
Now the aim is to apply the previously discussed results to the geometry of the Heisenberg groups. These groups are very widely studied and there are many references, expository and otherwise, to geometry and analysis in these groups (see for instance [5]). Here we will cover the basics of the Heisenberg group together with those properties that will be relevant to the contents of this paper. The n th Heisenberg group, denoted H n , is a nilpotent Lie group whose background manifold is C nˆR and whose Lie algebra has a step 2 strati cation h n " V ' V where V has dimension n, V has dimension 1 and satis es rV , V s " V with all other brackets being trivial. The most common way to represent H n is as the set C nˆR or R nˆR . We use this last two presentations interchangeably by identifying C n with R n in the usual way (i.e. z " x`iy). The group law on H n is then given by pz, tq˚pw, sq " pz`w, t`s´ ωpz, wqq.
With this representation, it is easy to see that H n is an R-bundle over C n with bundle map π : H n → C n given by πpz, tq " z. The left invariant vector elds are given by The rst n vector elds form a basis for a bracket-generating sub-bundle of h n . This allows us to use these vectors to create a well-de ned distance in H n . We say a C curve γ : I → H n is horizontal, or admissible, if γ pτq P SpantX , . . . , X n u, @τ P I. That is to say, if γ is horizontal then γ pτq " ř n j" a j pτqX j , hence we can de ne the length of γ as |γ| " The associated path distance, known as Carnot-Carathéodory distance, is given by dccpp, qq " inft|γ| : γ P C pr , s, H n q is horizontal, with γp q " p and γp q " qu.
This makes H n a sub-Riemannian manifold whose Hausdor dimension is di erent from its topological dimension. In fact, H n is Ahlfors p n` q-regular. More generally, for A Ă H n its Hausdor dimension with respect to the Heisenberg metric is greater than, or equal to, its dimension with respect to the Euclidean metric in R n` . In this section, dim A refers to the dimension with respect to the Heisenberg metric. To avoid confusion, if any reference to the Euclidean Hausdor dimension of a set is needed, it will be denoted by dim E A. The Heisenberg group also admits a gauge norm, known as the Korányi norm, which induces a metric that is bi-Lipschitz equivalent to dcc. Given that Hausdor dimension is invariant under bi-Lipschitz maps, and due to the explicit formula for the Korányi norm, the Korányi metric is often used in interchagebly with the Carnot-Caratheodory distance when exploring questions related to dimension of sets. The norm is given by ||pz, tq|| H " |z| ` t , and the metric, by where q´ is the group inverse of q (which happens to be´q). One of the most important properties of H n is that it admits a homogeneous structure given by the homogeneous dilations δrpz, tq " prz, r tq, r ą .
The map δr is an automorphism of the group that dilates distances by a factor of r. These dilations, combined with the group structure, make H n into a "almost vector space" of sorts. In fact, the Heisenberg group is the simplest example of a much larger class of groups, known as Carnot groups, which share many of these properties. With this homogeneous structure it makes sense to to talk about "vector subspaces", subgroups of H that are closed under homogeneous dilations. These are known as homogeneous subgroups and in H n they come in 2 types, those that intersect the t axis trivially and those that contain the whole t-axis. The former subgroups are called horizontal subgroups, while the latter are referred to as vertical subgroups. Suppose that V Ă H n is a horizontal subgroup. Then for pz, q, pw, q P V, their product pz, q˚pw, q " pz`w,´ ωpz, wqq must also be in V, so that ωpz, wq " . That is to say, V Ă C nˆt u must be contained in Vˆt u where V is an isotropic subspace of C n " R n . Indeed, there is a one to one correspondence between isotropic subspaces of R n and horizontal subgroups of H n . The notation V is used to denote the horizontal subgroup corresponding to the plane V. Moreover, if we denote by V K the Euclidean orthogonal complement of V, one can easily check that V K is a vertical subgroup of H n . For each V P G h p n, mq, H n admits a semi-direct splitting H n " V K¸V . This semi-direct splitting gives raise to well-de ned projection maps P V : H n → V and P V K : H n → V K de ned by "reading o " the respective component of a point p P H n . Horizontal projections are simply Euclidean orthogonal projections onto the corresponding isotropic subspace, whereas vertical projections are given by the formula P V K ppq " p˚P V ppq´ . The relation between horizontal subgroups and isotropic subspaces of H n allows to obtain strong results about Heisenberg dimension distortion by horizontal projections. A result along this lines was rst observed by Z.Balogh, E. Durand-Cartagena, K. Fässler, P. Mattila, and J.T. Tyson in [1], where they proved the following, This theorem is a consequence of Theorem 1.4 combined with the fact that the bundle map π : H n → C n does not increase Hausdor dimension of sets, and it decreases it by at most 2 (this is sharp). We can readily see how this, together with part (3) of Theorem 1.4, implies the following, Corollary 5.2. If s ą m` , IntpP V Aq ‰ ∅ for µ n,m -a.e. V P G h p n, mq.
Vertical projections, on the other hand, are not Euclidean projections. In fact, the best regularity of the map P V is (locally) -Holder. Therefore, analyzing dimension distortion by these maps is a much more complicated task that does not come as a corollary of any known projection theorems in Euclidean space. In [2], the authors obtained dimension distortion estimates for these vertical projections and conjectured what seems like feasible sharp bounds for said distortion. Later in [7] K. Fässler and R. Hovila improved the estimates for H , and recently T. Harris improved the H bounds even further in [9].
Here, with the results obtained for G h p n, mq one can easily prove Theorem 1.7 which in turned will help prove Theorem 1.9. Theorem 1.7. Let A, B Ă H n be Borel sets, and let V denote the horizontal subgroup corresponding to the isotropic plane V P G h p n, mq.
Proof. One can easily check that for any horizontal subgroup V, d H n t V " d E t V , therefore for any set D Ă H n dim P V D " dim E P V D. Moreover, P V " P V˝π and it was shown in [2] that dim E πpDq ě dim D´ . Therefore, the theorem is proven by simply applying Theorem 1.6 to the sets πpAq, πpBq Ă R nˆt u Ă H n .
Hidden within Theorem 1.7 is the following Lemma that will be useful later, Lemma 5.3. If s, σ P R are such that m` ă s, σ ď n` and µ, ν P MpH n q satisfy µpB H n pp, rqq À r s , and νpB H n pp, rqq À r σ , then for µ n,m´a lmost every V P G h p n, mq, P V# µ and P V# ν have densities µ V and ν V respectively and for a µ n,m´p ositive measure set of planes V P G h p n, mq, Proof. The fact that for almost every V P G h p n, mq P V# µ and P V# µ have densities is precisely the content of [2, Proposition 6.1]. As before, for each V, P V " P V˝π , so that P V# µ " P V# π # µ. Hence, the aim is to show that both I E m pπ # µq and I E m pπ # νq are nite, where I E m denotes the Euclidean m energy. The lemma will then follow from (2.4) and the computation in (3.1). It is enough to show that if β ą m` and η P MpH n q satis es ηpB H n pp, rqq À r β for all p P H n and r ą , then I E m pπ # ηq ă 8. First note that by de nition, We will bound the inner integral independently of p and use the fact that ηpH n q ă 8 to bound the double integral. Since η is compactly supported, there is R ą such that sptpηq Ă B E p , R q. For each xed z P R n , tq P H n : |πpqq´z| ď ru is a cylinder of (Euclidean) radius r over the ball B n E pz, rq. Therefore tq P H n : |πpqq´z| ď ru Ă B n E pz, rqˆr´R , R s. Where the last line is nite because β´ ą m. The lemma follows.
The projection theorem also help us obtain our claimed results for exceptional sets for the slicing theorem in H n . Vertical subgroups are normal subgroups of H n , its right cosets V K˚p , for p P H n , form a partition of H n . Just as in the Euclidean case, given a measure µ P MpH n q one can de ne the sliced measure µ V K˚p for each V and p P V. A detailed construction of these sliced measures can be found in [2], but it follows the same scheme as the construction of sliced measures by planes in Euclidean space. Here we will simply state the properties that will be relevant to us. Firstly, as expected from "sliced" measures, sptpµ V K˚p q Ă sptpµq X V K˚p . One can, however, say a lot more than that. In fact, for any Borel set Ω Ă V one has ż Ω µ V K˚p pH n qdH m ppq ď µpP´ V Ωq, (5.1) with equality whenever P V# µ ! H m . In particular, in this case if Ω " V we get ż V µ V K˚p pH n qdH m ppq " µpH n q.
Hence we also have P V# µppq " µ V K˚p pH n q. In [2], the authors proved the Heisenberg group analogue of Theorem 4.3. This will be used in the proof of Theorem 1.9 so it is stated here without proof.
Theorem 5.4. Let A Ă H n be a Borel set with ă H s pAq ă 8 for some m` ă s ď n` . Then, dimrA X pV K˚p qs " s´m, for H sˆµ n,m´a .e. px, Vq P AˆG h p n, mq.
The argument used to prove Theorem 4.3 is just a simple modi cation of the argument used in [2] to prove Theorem 5.4. In particular, as a consequence of the proof of Theorem 5.4 one obtains the analogue of Lemma 4.2.
Lemma 5.5. Let A Ă H n be a Borel set with ă H s pAq ă 8 for some m` ă s ď n` . Then, for all p P H n dimrA X pV K˚p qs ď s´m for µ n,m´a .e. V P G h p n, mq.
Just as before with Lemma 4.2, Lemma 5.5 holds at the level of dimension. One might expect that the stronger statement, at the level of measures, holds true. The following weaker result, analogous to Theorem 4.4, holds.
Theorem 5.6. Let m` ă s ď n. If A Ă H n is a Borel set such that ă H s pAq ă 8, then for almost every p P H n H s´m rA X pV K˚p qs ď s´m for µ n,m´a .e. V P G h p n, mq.
Proof. First, we show that for any R ą , for L n` ´almost every p P H n H s´m rA X pV K˚p qs1 N E pV,Rq ppq ă 8, for µ n,m´a lmost every V P G h p n, mq. Here, as before, N E pV, Rq is the Euclidean tubular neighborhood of V of radius R and L n` is the p n` q-dimensional Lebesgue measure. We remark that there is a constant d " dpnq, depending only on n, such that L n` " dH n` . So using L n` is equivalent to using the, arguably more intrinsic, measure H n` . To show (5.4), let λ " L n` t N E pV,Rq . Since N E pV, Rq " B V K ,E p , RqˆV, for any measurable set S Ă V we have P V# λpSq " λpP´ V pSqq " L n` pP´ V pSq X N E pV, Rqq " L n` ´m pB V K ,E p , RqqL m pSq "CpR, n, mqH m pSq.
HereCpR, n, mq is a constant, depending only on R, n, and m, which involves the L n` ´m´v olume of the ball of radius R in R n` ´m , as well as the universal constant c such that cH m " L m in R m . For each V, the map P V is 1-Lipschitz, so we can apply Elienberg's lemma to get