A survey on the Hausdorff dimension of intersections

Let $A$ and $B$ be Borel subsets of the Euclidean $n$-space with $\dim A + \dim B>n$. This is a survey on the question: what can we say about the Hausdorff dimension of the intersections $A\cap (g(B)+z)$ for generic orthogonal transformations $g$ and translations by $z$.


Introduction
1 The books [M5] and [M6] contain most of the required background information and the proofs of some of the results discussed below.
Let L n stand for the Lebesgue measure on the Euclidean n-space R n and let dim stand for the Hausdorff dimension and H s for s-dimensional Hausdorff measure. For A ⊂ R n , denote by M(A) the set of Borel measures µ with 0 < µ(A) < ∞ and with the compact support spt µ ⊂ A.
We let O(n) denote the orthogonal group of R n and θ n its Haar probability measure. The main fact needed about the measure θ n is the inequality: (1.1) θ n ({g ∈ O(n) : |x − g(z)| < r}) (r/|z|) n−1 for x, z ∈ R n , r > 0.
This is quite easy, in fact trivial in the plane. Let A and B be Borel subsets of R n with Hausdorff dimensions s = dim A and t = dim B. What can we say about the Hausdorff dimensions of the intersections of A and typical rigid motions of B? More precisely, of dim A ∩ (g(B) + z) for almost all g ∈ O(n) and for z ∈ R n in a set of positive Lebesgue measure. Optimally one could hope that this dimension is given by the bigger of the numbers s + t − n and 0, which happens when smooth surfaces meet in a general position.
The problem on the upper bound is much easier than on the lower bound. Let (1.2) V z = {(x, y) ∈ R n × R n : x = y + z}, z ∈ R n , be the z translate of the diagonal in R n ×R n , and let π be the projection π(x, y) = x. Then (1.3) A ∩ (g(B) + z) = π((A × g(B)) ∩ V z ), and it follows from a Fubini-type inequality for Hausdorff dimension, [M5,Theorem 7.7], that for any g ∈ O(n), Even the weaker condition that the Hausdorff and packing dimensions of A agree suffices, see [M5], pp. 115-116. Then we have Without some extra condition this inequality fails badly: for any 0 ≤ s ≤ n there exists a Borel set A ⊂ R n of dimension s such that dim A ∩ f (A) = s for all similarity maps f of R n . This was proved by Falconer in [F3], see also Example 13.19 in [M5] and the further references given there. We have the lower bound for the dimension of intersections if we use larger transformation groups, for example similarities: Theorem 1.1. Let A and B be Borel subsets of R n with dim A + dim B > n. Then for every ε > 0, for almost all g ∈ O(n) and almost all r > 0.
If A and B have positive and finite Hausdorff measure, ε is not needed. This theorem was proved in the 1980s independently by Kahane [K] and in [M2]. More generally, Kahane proved that the similarities can be replaced by any closed subgroup of the general linear group of R n which is transitive outside the origin. He gave applications to multiple points of stochastic processes.
There are many special cases where the equality dim A ∩ (g(B) + z) = dim A + dim B − n holds for almost all g and for z in a set of positive measure. The case where one of the sets is a plane, initiated by Marstrand in [M], has been studied a lot, see discussions in [M5,Chapter 10] and [M6,Chapter 6], and [MO] for a more recent result. More generally, one of the sets can be rectifiable, see [M2].
The main open problem is: what conditions on the Hausdorff dimensions or measures of A and B guarantee that for θ n almost all g ∈ O(n), or perhaps for all ε > 0, If one of the sets is a Salem set, that is, it supports a measure with an optimal Fourier decay allowed by its Hausdorff dimension, then (1.8) holds without dimensional restrictions, see [M4]. I expect (1.8) to be true for all Borel subsets A and B of R n . Below I shall discuss some partial results on this question. I shall also say something about the exceptional sets of transformations.
In this survey I shall concentrate on Hausdorff dimension and general Borel sets. For remarks and references about related results on other dimensions, see [M5,Section 13.20] and [M6,Section 7.3]. There is a rich literature on various questions about intersections of dynamically generated and related sets. For recent results and further references, see [S1], [Wu], [Y]. For probabilistic sets, see [SS] and its references.
I would like to thank the referees for useful comments.

Projections and plane intersections
This topic can be thought of as a study of integral-geometric properties of fractal sets and Hausdorff dimension. Let us briefly review some of the basic related results on projections and plane sections. This was started by Marstrand in [M] in the plane. His main results in general dimensions are the following. Let G(n, m) be the Grassmannian of linear m-dimensional subspaces of R n and P V : R n → V the orthogonal projection onto V ∈ G(n, m). Let also γ n,m be the orthogonally invariant Borel probability measure on G(n, m).
Theorem 2.1. Let A ⊂ R n be a Borel set. Then for almost all V ∈ G(n, m), Theorem 2.2. Let n − m ≤ s ≤ n and let A ⊂ R n be H s measurable with 0 < H s (A) < ∞. Then for almost all V ∈ G(n, m), and for almost all V ∈ G(n, n − m) and for H s almost all x ∈ A, One can sharpen these results by deriving estimates on the Hausdorff dimension of the exceptional sets of the planes V . For the first part of Theorem 2.1 this was first done by Kaufman in [Ka] in the plane, then in [M1] and [KM] in higher dimensions. For the second part of Theorem 2.1 the exceptional set estimates were proven by Falconer in [F1]. Thus we have, recall that dim G(n, m) = m(n − m): Theorem 2.3. Let A ⊂ R n be a Borel set with s = dim A. Then These inequalities are sharp by the examples in [KM] (and their modifications), but the proof for (2.5) also gives the upper bound t − m + m(n − m) if dim A on the left hand side is replaced by t, 0 ≤ t ≤ dim A. Then for t < dim A this is not always sharp, see the discussion in [M6,Section 5.4].
For the plane sections Orponen proved in [Or1], see also [M6,Theorem 6.7], the exceptional set estimate (which of course is sharp, as (2.6) is): We can also ask for exceptional set estimates corresponding to (2.4). We proved with Orponen [MO] the following: Very likely, the bound n − m is not sharp. When m = 1, probably the sharp bound should be 2(n − 1) − s in accordance with Orponen's sharp result for radial projections in [Or2].
Another open question is whether there could be some sort of non-trivial estimate for the dimension of the exceptional pairs (x, V ).

Some words about the methods
The methods in all cases use Frostman measures. Suppose that the Hausdorff measures H s (A) and H t (B) are positive. Then there are µ ∈ M(A) and ν ∈ M(B) such that µ(B(x, r)) ≤ r s and ν(B(x, r)) ≤ r t for x ∈ R n , r > 0. In particular, for 0 < s < dim A and 0 < t < dim B there are µ ∈ M(A) and ν ∈ M(B) such that I s (µ) < ∞ and I t (ν) < ∞, where the s energy I s (µ) is defined by Then the goal is to find intersection measures λ g,z ∈ M(A ∩ (g(B) + z)) such that This would give (1.8).
There are two closely related methods to produce these measures. The first, used in [M2], is based on (1.3): the intersections A ∩ (g(B) + z) can be realized as level sets of the projections S g : For this to work, one needs to know that This is usually proved by establishing the L 2 estimate which, by Plancherel's formula, is equivalent to where F stands for the Fourier transform. The second method, used in [K], is based on convolution approximation. Letting ψ ε , ε > 0, be a standard approximate identity, set ν ε = ψ ε * ν and Then the plan is to show that when ε → 0, the measures ν g,z,ε µ converge weakly to the desired intersection measures. No Fourier transform is needed to prove Theorem 1.1, but the proofs of all theorems discussed below, except Theorems 6.1, 6.2, 6.3 and 6.4, rely on the Fourier transform defined by The basic reason for its usefulness in this connection is the formula which is a consequence of Parseval's formula and the fact that the distributional Fourier transform of the Riesz kernel k s , k s (x) = |x| −s , is a constant multiple of k n−s .
The decay of the spherical averages, is the surface measure on the sphere S(r) = {x ∈ R n : |x| = r}, often plays an important role. By integration in polar coordinates, if σ(µ)(r) r −t for r > 0 and for some t > 0, then I s (µ) < ∞ for 0 < s < t. Hence the best decay we can hope for under the finite s energy assumption (or the Frostman assumption µ(B(x, r)) ≤ r s )) is r −s . This is true when s ≤ (n − 1)/2, see [M6,Lemma 3.5], but false otherwise.
The decay estimates for σ(µ)(r) have been studied by many people, discussion can be found in [M6,Chapter 15]. The best known estimates, due to Wolff, [W], when n = 2 (the proof can also be found in [M6,Chapter 16]) and to Du and Zhang, [DZ], in the general case, are the following: The factor µ(B(0, 1)) can be checked from the proofs in [W], [DZ] and [M6]. The essential case for the first estimate is s > n/2, otherwise the second and third are better. Up to ε these estimates are sharp when n = 2. When n ≥ 3 the sharp bounds are not known for all s, see [D] for discussion and the most recent examples. As mentioned above, the last bound is always sharp.

The first theorem
If one of the sets has dimension bigger than (n + 1)/2 we have the following theorem. It was proved in [M3], see also [M5,Theorem 13.11] or [M6,Theorem 7.4

]:
Theorem 4.1. Let s and t be positive numbers with s + t > n and s > (n + 1)/2. Let A and B be Borel subsets of R n with H s (A) > 0 and H t (B) > 0. Then The proof is based on the slicing method. The key estimate is if µ ∈ M(R n ), 0 < δ ≤ r and (n + 1)/2 ≤ s < n. This is combined with the inequality (1.1). The inequality (4.2) is obtained with the help of the Fourier transform, and that is the only place in the proof of Theorem 4.1 where the Fourier transform is needed.
One problem of extending Theorem 4.1 below the dimension bound (n + 1)/2 is that the estimate (4.2) then fails, at least in the plane by [M6,Example 4.9] and in R 3 by [IS].
In Section 7 we discuss estimates on the exceptional sets of orthogonal transformations. The proof of Theorem 7.1 gives another proof for Theorem 4.1 but under the stronger assumption s + t > n + 1. On the other hand, Theorem 6.5 below holds with the assumption s + (n − 1)t/n > n but under the additional condition of positive lower density. Of course, s + (n − 1)t/n > n is sometimes stronger and sometimes weaker than s > (n + 1)/2, s + t > n. For example, consider these in the plane. When s = t, the first one says s > 4/3 and the second one s > 3/2. On the other hand, when s is slightly bigger than 3/2, the first requires t to be at least about 1, but the second allows t = 1/2. Theorem 4.1 says nothing in R 1 , and there is nothing to say: in [M2] I constructed compact sets A, B ⊂ R such that dim A = dim B = 1 and A ∩ (B + z) contains at most one point for any z ∈ R. With A, B ⊂ R as above, the n-fold Cartesian products A × · · · × A ⊂ R n and B × · · · × B ⊂ R n yield the corresponding examples in R n , that is, just with translations we get nothing in general.
Donoven and Falconer proved in [DF] an analogue of Theorem 4.1 for the isometries of the Cantor space. They didn't need any dimensional restrictions. They used martingales to construct the desired random measures with finite energy integrals on the intersections.

The projections S g
We now discuss a bit more the projections S g , recall (3.4). They are particular cases of restricted projections, which recently have been studied extensively, see [M6,Section 5.4], [M9], [H2] and [GGGHMW] and the references given there. Restricted means that we are considering a lower dimensional subspace of the Grassmannian G(2n, n). For the full Grassmannian we have Marstrand's projection theorem 2.1.
As mentioned above, to prove Theorem 4.1 one first needs to know (3.6) when s + t > n and s > (n + 1)/2 and µ and ν have finite s and t energies. A simple proof using spherical averages is given in [M6,Lemma 7.1]. This immediately yields the weaker result: with the assumptions of Theorem 4.1, for almost all g ∈ O(n), (5.1) L n ({z ∈ R n : A ∩ (g(B) + z) = ∅}) > 0, because (5.1) is equivalent to L n (S g (A × B)) > 0. Even for this I don't know if the assumption s > (n + 1)/2 is needed. Let us first look at general Borel subsets of R 2n : Theorem 5.1. Let A ⊂ R 2n be a Borel set. If dim A > n + 1, then L n (S g (A)) > 0 for θ n almost all g ∈ O(n).
This was proved in [M9]. That paper also contains dimension estimates for S g (A) when dim A ≤ n + 1 and estimates on the dimension of exceptional sets of transformations g. In particular, if n ≤ dim A ≤ n + 1, then The bound n + 1 in Theorem 5.1 is sharp. This was shown by Harris in [H1]. First, (5.2) is sharp. The example for dim A = n is simply the diagonal D = {(x, x) : x ∈ R n }. To see this suppose that g ∈ O(n) is such that det g = (−1) n+1 , which is satisfied by half of the orthogonal transformations. Then by some linear algebra g has a fixed point, whence the kernel of x → S g (x, x) is non-trivial, so dim S g (D) ≤ n − 1. Taking the Cartesian product of D with a one-dimensional set of zero H 1 measure, we obtain A with dim A = n + 1 and L n (S g (A)) = 0, which proves the sharpness.
But this only gives an example A of dimension n + 1 for which L n (S g (A)) = 0 for g ∈ O(n) with measure 1/2. Is there a counter-example that works for almost all g ∈ O(n)?
Here are the basic ingredients of the proof of Theorem 5.1. They were inspired by Oberlin's paper [O].
Let 0 < n+1 < s < dim A and µ ∈ M(A) with I s (µ) < ∞, and let µ g ∈ M(S g (A)) be the push-forward of µ under S g . The Fourier transform of µ g is given by By fairly standard arguments, using also the inequality (1.1), one can then show that for R > 1, This is summed over the dyadic annuli, R = 2 k , k = 1, 2, . . . . The sum converges since s > n + 1. Hence for θ n almost all g ∈ O(n), µ g is absolutely continuous with L 2 density, and so L n (S g (A)) > 0.
For product sets we can improve this, which is essential for the applications to intersections: Theorem 5.2. Let A, B ⊂ R n be Borel sets. If dim A + (n − 1) dim B/n > n or dim A + dim B > n and dim A > (n + 1)/2, then L n (S g (A × B)) > 0 for θ n almost all g ∈ O(n).
The case dim A > (n + 1)/2 is a special case of Theorem 4.1, recall (5.1). The proof of the case dim A + (n − 1) dim B/n > n is based on the spherical averages and the first estimate of (3.11). Here is a sketch.

Level sets and intersections
The following results with positive lower density assumptions were proven in [M8]. It turned out that those assumptions can be removed quite easily, as explained below.
The estimate (3.7) can be used to derive information on the Hausdorff dimension of the level sets of S g , and hence, by (3.5), of intersections. We shall first discuss a more general version of this principle: a quantitative projection theorem leads to estimates of the Hausdorff dimension of level sets. This is also how in [M5,Chapter 10] the proof for Marstrand's section theorem 2.2 runs.
We consider the following general setting. Let P λ : R n → R m , λ ∈ Λ, be orthogonal projections, where Λ is a compact metric space. Suppose that λ → P λ x is continuous for every x ∈ R n . Let also ω be a finite non-zero Borel measure on Λ. We denote by D(µ, ·) the Radon-Nikodym derivative of a measure µ on R m . Theorem 6.1. Let s > m. Suppose that there exists a positive number C such that P λ♯ µ ≪ L m for ω almost all λ ∈ Λ and for H s almost all x ∈ A, then for ω almost all λ ∈ Λ, (6.2) L m ({u ∈ R m : dim P −1 λ {u} ∩ A = s − m}) > 0. Theorem 6.2. Let s > m. Suppose that there exists a positive number C such that P λ♯ µ ≪ L m for ω almost all λ ∈ Λ and (6.3) D(P λ♯ µ, u) 2 dL m u dωλ < Cµ(B n (0, 1)) whenever µ ∈ M(B n (0, 1)) is such that µ(B(x, r)) ≤ r s for x ∈ R n , r > 0. If A ⊂ R n is H s measurable, 0 < H s (A) < ∞, then for ω almost all λ ∈ Λ, For an application to intersections we shall need the following product set version of Theorem 6.2, Theorem 6.4. There P λ : R n × R p → R m , λ ∈ Λ, m < n + p, are orthogonal projections with the same assumptions as before.
I give a few words about the proofs of Theorems 6.1 and 6.2. First, notice that D(P λ♯ (µ), u) is given by Let µ be the restriction of H s to a subset of A so that µ satisfies the Frostman s condition. Then (6.1) is applied to the measures µ a,r,δ = r −s T a,r♯ (µ δ B(a, r)) ∈ M(B(0, 1)), a ∈ R n , r > 0, δ > 0, where µ δ (B) = δ −n B µ(B(x, r)) dL n x, T a,r (x) = (x − a)/r is the blow-up map and µ δ B(a, r) is the restriction of µ δ to B(a, r). This leads for almost all x ∈ A, λ ∈ Λ, to (6.9) lim which is a Frostman type condition along the level sets of the P λ . With some further work it leads to (6.2). The proof of Theorem 6.3 is similar. The proof of Theorem 6.2 is the same as that of Theorem 6.1 observing that the assumption of positive lower density is only used to get the last line of page 394 in [M8], and this, with µ(B j,i ) replaced by µ(2B j,i ), now follows from the assumption (6.3). Here µ(2B j,i ) is fine, since a few lines earlier in [M8] we can require that the balls 2B j,i too have bounded overlap.
Theorem 6.4 together with the quantitative version (5.4) of Theorem 5.2 and with (3.5) can be applied to the projections S g to obtain the following result on the Hausdorff dimension of intersections: Theorem 6.5. Let s, t > 0 with s+(n−1)t/n > n and let A ⊂ R n be H s measurable with 0 < H s (A) < ∞, and let B ⊂ R n be H t measurable with 0 < H t (B) < ∞. Then for θ n almost all g ∈ O(n), Harris [H2] proved the analogue of the part (2.2) of Theorem 2.1 for vertical projections in the Heisenberg group H 1 . He applied this with a method partially similar to that of the proof of Theorem 6.2 to get the analogue of the intersection theorem 2.2. This lead me to remove the lower density assumptions.

Exceptional set estimates
Recall the exceptional set estimates for orthogonal projections and for intersections with planes from Chapter 2. Now we discuss some similar results from [M7] for intersections.
First we have an exceptional set estimate related to Theorem 4.1. But we need a bit stronger assumption: the sum of the dimensions is required to be bigger than n+1, rather than just one of the sets having dimension bigger than (n+1)/2. Recall that the dimension of O(n) is n(n − 1)/2.
In the case where one of the sets has small dimension we have the following improvement of Theorem 7.1: Theorem 7.2. Let A and B be Borel subsets of R n and suppose that dim A ≤ The last decay estimate in (3.11) of spherical averages is essential for the proof. The reason why the assumption dim A ≤ (n−1)/2 leads to a better result is that that estimate in (3.11) is stronger than the others. For dim A > (n−1)/2 the inequalities (3.11) would only give weaker results with u replaced by a smaller number, see [M7,Section 4].
If one of the sets supports a measure with sufficiently fast decay of the averages σ(µ)(r), we can improve the estimate of Theorem 7.1. Then the results even hold without any rotations provided the dimensions are big enough. In particular, we have the following result in case one of the sets is a Salem set. By definition, A is a Salem set if for every 0 < s < dim A there is µ ∈ M(A) such that | µ(x)| 2 |x| −s . A discussion on Salem sets can be found, for example, in [M6], Section 3.6.
Theorem 7.3. Let A and B be Borel subsets of R n and suppose that A is a Salem set. Suppose that 0 < u < dim A + dim B − n.
Could this hold for general sets, perhaps in the form that dim E = 0, if dim A + dim B > 2n − 1? It follows from Theorem 2.4 that this is true if one of the sets is a plane. In R 2 a slightly stronger question reads: if s + t > 2 and A and B are Borel subsets of R 2 with H s (A) > 0 and H t (B) > 0, is there E ⊂ O(2) with dim E = 0, if s + t ≥ 3, and dim E ≤ 3 − s − t, if s + t ≤ 3, such that for g ∈ O(2) \ E, L 2 ({z ∈ R 2 : dim A ∩ (g(B) + z) ≥ s + t − 2}) > 0? 8. Some relations to the distance set problem There are some connections of this topic to Falconer's distance set problem. For general discussion and references, see for example [M6]. Falconer showed in [F2] that for a Borel set A ⊂ R n the distance set {|x − y| : x, y ∈ A} has positive Lebesgue measure if dim A > (n + 1)/2. We had the same condition in Theorem 4.1. Also for distance sets it is expected that dim A > n/2 should be enough.
When n = 2 Wolff [W] improved 3/2 to 4/3 using (3.11). Observe that when dim A = dim B, the assumption dim A + dim B/2 > 2 in Theorem 6.5 becomes dim A > 4/3 and it is the same as Wolff's. This is no coincidence: both results use Wolff's estimate (3.11).
The proofs of distance set results often involve the distance measure δ(µ) of a measure µ defined by δ(µ)(B) = µ × µ({(x, y) : |x − y| ∈ B}), B ⊂ R. The crucial estimate (4.2) means that δ(µ) is absolutely continuous with bounded density if I (n+1)/2 (µ) < ∞. Hence it yields Falconer's result. As mentioned before we cannot hope to get bounded density when s < (n+1)/2, at least when n = 2 or 3. In many of the later improvements one verifies absolute continuity with L 2 density. For example, Wolff showed that δ(µ) ∈ L 2 (R), if I s (µ) < ∞ for some s > 4/3. To do this he used decay estimates for the spherical averages σ(µ)(r) and proved (3.11) for n = 2. The proofs of the most recent, and so far the best known, distance set results in [DZ], [GIOW], [DGOWWZ] and [DIOWZ] are quite involved using deep harmonic analysis techniques; restriction and decoupling. In the plane the result of [GIOW]