1. Introduction
The books [
1,
2] contain most of the required background information and the proofs of some of the results discussed below.
Let stand for the Lebesgue measure on the Euclidean n-space and let dim stand for the Hausdorff dimension and for the s-dimensional Hausdorff measure. For , denote by the set of Borel measures with and with the compact support .
We let
denote the orthogonal group of
and
its Haar probability measure. The main fact needed about the measure
is the inequality
This is quite easy, and is in fact trivial in the plane.
Let A and B be Borel subsets of with Hausdorff dimensions and . What can we say about the Hausdorff dimensions of the intersections of A and typical rigid motions of B—more precisely, of for almost all and for in a set of positive Lebesgue measure? Optimally, one could hope that this dimension is given by the larger of the numbers 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
be the
z translate of the diagonal in
, and let
be the projection
. Then,
and it follows from a Fubini-type inequality for the Hausdorff dimension [
1] (Theorem 7.7) that for any
,
provided
. We have always
and the equation
holds if, for example,
, and one of the sets has positive lower density, say
Even the weaker condition that the Hausdorff and packing dimensions of
A agree suffices; see [
1], pp. 115–116. Then, we have
provided
. Without some extra condition, this inequality fails seriously: for any
, there exists a Borel set
of dimension
s such that
for all similarity maps
f of
. This was proven by Falconer in [
3]; see also Example 13.19 in [
1] 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. Let A and B be Borel subsets of with . Then, for every ,for almost all and almost all . If
A and
B have positive and finite Hausdorff measures,
is not needed. This theorem was proven in the 1980s independently by Kahane [
4] and in [
5]. More generally, Kahane proved that the similarities can be replaced by any closed subgroup of the general linear group of
that is transitive outside the origin. He gave applications to multiple points of stochastic processes.
There are many special cases where the equality
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 [
6], has been studied a lot; see discussions in [
1] (Chapter 10) and [
2] (Chapter 6), and [
7] for a more recent result. More generally, one of the sets can be rectifiable; see [
5].
The main open problem is as follows: what conditions on the Hausdorff dimensions or measures of
A and
B guarantee that for
almost all
,
or perhaps for all
,
If one of the sets is a Salem set, i.e., it supports a measure with an optimal Fourier decay allowed by its Hausdorff dimension, then (
8) holds without dimensional restrictions; see [
8]. I expect (
8) to be true for all Borel subsets
A and
B of
.
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 the Hausdorff dimension and general Borel sets. For remarks and references about related results on other dimensions, see [
1] (Section 13.20) and [
2] (Section 7.3). There is a rich body of literature on various questions about intersections of dynamically generated and related sets. For recent results and further references, see [
9,
10,
11]. For probabilistic sets, see [
12] and its references.
I would like to thank the referees for their useful comments.
2. Projections and Plane Intersections
This topic can be thought of as a study of the integral-geometric properties of fractal sets and the Hausdorff dimension. Let us briefly review some of the basic related results on projections and plane sections. This was started by Marstrand in [
6] in the plane. His main results in general dimensions are the following. Let
be the Grassmannian of linear
m-dimensional subspaces of
and
the orthogonal projection onto
. Let also
be the orthogonally invariant Borel probability measure on
.
Theorem 2. Let be a Borel set. Then, for almost all ,and Theorem 3. Let and let be measurable with . Then, for almost all ,and for almost all and for almost all , 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, this was first done by Kaufman in [
13] in the plane, and then in [
14,
15] in higher dimensions. For the second part of Theorem 2, the exceptional set estimates were proven by Falconer in [
16]. Thus we have, recall that
.
Theorem 4. Let be a Borel set with . Then,and These inequalities are sharp by the examples in [
14] (and their modifications), but the proof for (
13) also gives the upper bound
if
on the left-hand side is replaced by
. Then, for
, this is not always sharp; see the discussion in [
2] (Section 5.4).
For the plane sections, Orponen proved in [
17]—see also [
2] (Theorem 6.7)—the exceptional set estimate (which of course is sharp, as (
14) is).
Theorem 5. Let and let be measurable with . Then, there is a Borel set such thatand for We can also ask for exceptional set estimates corresponding to (
12). We proved with Orponen [
7] the following:
Theorem 6. Let and let be measurable with . Then, the set B of points withhas dimension . Very likely, the bound
is not sharp. When
, probably, the sharp bound should be
in accordance with Orponen’s sharp result for radial projections in [
18].
Another open question is whether there could be some sort of non-trivial estimate for the dimension of the exceptional pairs .
3. Some Words about the Methods
The methods in all cases use Frostman measures. Suppose that the Hausdorff measures
and
are positive. Then, there are
and
such that
and
for
. In particular, for
and
, there are
and
such that
and
, where the
s energy
is defined by
Then, the goal is to find intersection measures
such that
There are two closely related methods to produce these measures. The first, used in [
5], is based on (
3): the intersections
can be realized as level sets of the projections
:
Notice that the map is essentially the orthogonal projection onto the n-plane .
Thus, one slices (disintegrates)
(
is the push-forward) with the planes
For this to work, one needs to know that
This is usually proven by establishing the
estimate
which, by Plancherel’s formula, is equivalent to
where
stands for the Fourier transform.
The second method, used in [
4], is based on convolution approximation. Letting
, be a standard approximate identity, set
and
Then, the plan is to show that when , the measures converge weakly to the desired intersection measures.
No Fourier transform is needed to prove Theorem 1, but the proofs of all theorems discussed below, except Theorems 10 and 11, 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
, is a constant multiple of
.
The decay of the spherical averages,
of
, where
is the surface measure on the sphere
, often plays an important role. By integration in polar coordinates, if
for
and for some
, then
for
. Hence, the best decay that we can hope for under the finite
s energy assumption (or the Frostman assumption
) is
. This is true when
—see [
2] (Lemma 3.5)—but false otherwise.
The decay estimates for
have been studied by many people; a discussion can be found in [
2] (Chapter 15). The best-known estimates, due to Wolff [
19] when
(the proof can also be found in [
2] (Chapter 16)) and to Du and Zhang [
20] in the general case, are the following. Let
with
for
. Then, for all
,
The essential case for the first estimate is
; otherwise, the second and third are better. Up to
, these estimates are sharp when
. When
, the sharp bounds are not known for all
s; see [
21] for a discussion and the most recent examples. As mentioned above, the last bound is always sharp.
4. The First Theorem
If one of the sets has a dimension greater than
, we have the following theorem. It was proven in [
22]; see also [
1] (Theorem 13.11) or [
2] (Theorem 7.4).
Theorem 7. Let s and t be positive numbers with and . Let A and B be Borel subsets of with and . Then,for almost all . The proof is based on the slicing method. The key estimate is
if
and
. This is combined with the inequality (
1).
The inequality (
28) is obtained with the help of the Fourier transform, and that is the only place in the proof of Theorem 7 where the Fourier transform is needed.
One problem of extending Theorem 7 below the dimension bound
is that the estimate (
28) then fails, at least in the plane by [
2] (Example 4.9) and in
by [
23].
In
Section 7, we discuss estimates on the exceptional sets of orthogonal transformations. The proof of Theorem 13 gives another proof for Theorem 7 but under the stronger assumption
. On the other hand, Theorem 12 below holds with the assumption
but under the additional condition of positive lower density. Of course,
is sometimes stronger and sometimes weaker than
. For example, consider these in the plane. When
, the first one says
and the second one
. On the other hand, when
s is slightly larger than
, the first requires
t to be at least 1, but the second allows
.
Theorem 7 says nothing in
, and there is nothing to say: in [
5], I constructed compact sets
such that
and
contains at most one point for any
. With
as above, the
n-fold Cartesian products
and
yield the corresponding examples in
—that is, simply, with translations, we obtain nothing in general.
Donoven and Falconer proved in [
24] an analogue of Theorem 7 for the isometries of the Cantor space. They did not need any dimensional restrictions. They used martingales to construct the desired random measures with finite energy integrals on the intersections.
5. The Projections
We now discuss further the projections
; recall (
19). They are particular cases of restricted projections, which recently have been studied extensively; see [
2] (Section 5.4), and [
25,
26] and the references given there. Restricted means that we are considering a lower-dimensional subspace of the Grassmannian
. For the full Grassmannian, we have Marstrand’s projection Theorem 2.
As mentioned above, to prove Theorem 7, one first needs to know (
21) when
and
and
and
have finite
s and
t energies. A simple proof using spherical averages is given in [
2] (Lemma 7.1). This immediately yields the weaker result: with the assumptions of Theorem 7, for almost all
,
because (
29) is equivalent to
. Even for this, I do not know if the assumption
is needed.
Let us first look at general Borel subsets of .
Theorem 8. Let be a Borel set. If , then for almost all .
This was proven in [
26]. The paper also contains dimension estimates for
when
and estimates on the dimension of exceptional sets of transformations
g. In particular, if
, then
The bound
in Theorem 8 is sharp. This was shown by Harris in [
27]. First, (
30) is sharp. The example for
is simply the diagonal
. To see this, suppose that
is such that
, which is satisfied by half of the orthogonal transformations. Then, by some linear algebra,
g has a fixed point, whence the kernel of
is non-trivial, so
. Taking the Cartesian product of
D with a one-dimensional set of zero
measure, we obtain
A with
and
, which proves the sharpness.
However, this only gives an example A of dimension for which for with measure . Is there a counter-example that works for almost all ?
Here are the basic ingredients of the proof of Theorem 8. They were inspired by Oberlin’s paper [
28].
Let
and
with
, and let
be the push-forward of
under
. The Fourier transform of
is given by
By fairly standard arguments, using also the inequality (
1), one can then show that for
,
This is summed over the dyadic annuli, . The sum converges since . Hence, for almost all , is absolutely continuous with density, and so .
For product sets, we can improve this, which is essential for the applications to intersections.
Theorem 9. Let be Borel sets. If or and , then for almost all .
The case
is a special case of Theorem 7; recall (
29). The proof of the case
is based on the spherical averages and the first estimate of (
26). Here is a sketch.
Let
and
such that
, and let
with
for
. Let
Then,
. By (
26), we have
This gives Theorem 9.
In fact, for some results on the intersections below, we again need absolute continuity as in (
21). In the case
, we need the quantitative estimate: if
and
for
, then
with the implicit constant independent of
and
. The arguments described above give this too.
6. Level Sets and Intersections
The estimate (
33) can be used to derive information on the Hausdorff dimension of the level sets of
, and hence, by (
20), of intersections. The following results were proven in [
29]. 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 [
1] (Chapter 10), the proof for Marstrand’s section Theorem 3 runs.
We consider the following general setting. Let be orthogonal projections, where is a compact metric space. Suppose that is continuous for every . Let also be a finite non-zero Borel measure on . We denote by the Radon–Nikodym derivative of a measure on .
Theorem 10. Let . Suppose that there exists a positive number C such that for ω almost all andwhenever is such that for . If is measurable, and (recall (5)) for almost all , then for ω almost all , For an application to intersections, we shall need the following product set version of Theorem 10. There, are orthogonal projections with the same assumptions as before.
Theorem 11. Let with . Suppose that there exists a positive number C such that for ω almost all andwhenever are such that for , and for . If is measurable, , is measurable, , for almost all , and for almost all , then for ω almost all , I do not know if the assumptions on positive lower density are needed.
I give a few words about the proof of Theorem 10. First, notice that
is given by
Let
be the restriction of
to a subset of
A so that
satisfies the Frostman
s condition. Then, (
34) is applied to the measures
where
,
is the blow-up map and
is the restriction of
to
. This leads for almost all
, to
which is a Frostman-type condition along the level sets of the
. With some further work, it leads to (
35). The proof of Theorem 11 is similar.
Theorem 11, together with the quantitative version of Theorem 9 and with (
20), can be applied to the projections
to obtain the following result on the Hausdorff dimension of intersections.
Theorem 12. Let with and let be measurable with , and let be measurable with . Suppose that for almost all and for almost all . Then, for almost all , Again, I do not know if the positive lower density assumptions are needed for the lower bound . As mentioned before, they are needed for the upper bound.
7. Exceptional Set Estimates
Recall the exceptional set estimates for orthogonal projections and for intersections with planes from
Section 2. Now, we discuss some similar results from [
30] for intersections.
First, we have an exceptional set estimate related to Theorem 7. However, we need a slightly stronger assumption: the sum of the dimensions is required to be larger than , rather than only one of the sets having a dimension larger than . Recall that the dimension of is .
Theorem 13. Let s and t be positive numbers with . Let A and B be Borel subsets of with and . Then, there is such thatand for , The proof is based on the Fourier transform and the convolution approximation mentioned in
Section 3. Instead of
, one uses a Frostman measure
on the exceptional set
E: if
is such that
for all
and
, then for
,
This replaces the inequality (
1).
In the case where one of the sets has a small dimension, we have the following improvement of Theorem 13.
Theorem 14. Let A and B be Borel subsets of and suppose that . If , then there is withsuch that for , The last decay estimate in (
26) of spherical averages is essential for the proof. The reason that the assumption
leads to a better result is that this estimate in (
26) is stronger than the others. For
, the inequalities (
26) would only give weaker results with
u replaced by a smaller number; see [
30] (Section 4).
If one of the sets supports a measure with sufficiently fast decay of the averages
, we can improve the estimate of Theorem 13. Then, the results even hold without any rotations, provided that the dimensions are large enough. In particular, we have the following result in the event that one of the sets is a Salem set. By definition,
A is a Salem set if, for every
, there is
such that
. A discussion on Salem sets can be found, for example, in [
2], Section 3.6.
Theorem 15. Let A and B be Borel subsets of and suppose that A is a Salem set. Suppose that .
(
a)
If , then(
b)
If , then there is withsuch that for , Could this hold for general sets, perhaps in the form that
, if
? It follows from Theorem 5 that this is true if one of the sets is a plane. In
, a slightly stronger question reads as follows: if
and
A and
B are Borel subsets of
with
and
, is there
with
, if
, and
, if
, such that for
,
8. Some Relations to the Distance Set Problem
There are some connections of this topic to Falconer’s distance set problem. For a general discussion and references, see, for example, [
2]. Falconer showed in [
31] that, for a Borel set
, the distance set
has a positive Lebesgue measure if
. We had the same condition in Theorem 7. Moreover, for distance sets, it is expected that
should be enough.
When
, Wolff [
19] improved
to
using (
26). Observe that when
, the assumption
in Theorem 12 becomes
and it is the same as Wolff’s. This is no coincidence: both results use Wolff’s estimate (
26).
The proofs of distance set results often involve the distance measure
of a measure
defined by
The crucial estimate (
28) means that
is absolutely continuous with bounded density if
. Hence, it yields Falconer’s result. As mentioned before, we cannot hope to obtain bounded density when
, at least when
or 3. In many of the later improvements, one verifies absolute continuity with
density. For example, Wolff showed that
, if
for some
. To do this, he used decay estimates for the spherical averages
and proved (
26) for
. The proofs of the most recent, and so far the best known, distance set results in [
20,
32,
33,
34] involve using deep harmonic analysis techniques; restriction and decoupling. In the plane, the result of [
33] says that the distance set of
A has a positive Lebesgue measure if
. See Shmerkin’s survey [
35] for the distance set and related problems.
Distance measures are related to the projections
by the following:
at least if
and
are smooth functions with compact support; see [
26] (Section 5.2).
Since, by an example in [
33], when
, for any
,
is not enough for
to be in
, probably, because of (
45), it is not enough for
to be in
. However, in [
33], it was shown that if
for some
, there is a complex valued modification of
with good
behavior. In even higher dimensions, similar results were proven in [
34] with
in place of
. Could these methods be used to show, for instance, that if
and
, then
for almost all
?