After my previous post on a Fourier-analysis approach to Furstenberg’s problem, I would like to present another application of the Fourier transform of measures: a simple proof of Marstrand theorem about projections of Cantor sets. Before entering the details, let me tell you what are my motivations on this topic: besides the beauty of the Fourier-analysis approach to a relevant geometrical problem, this particular proof of Marstrand theorem was generalized by C. Moreira and J. C. Yoccoz in their work on stable intersections of regular Cantor sets so that it could be applied to the study of homoclinic tangencies (Dynamical Systems) and the Markov/Lagrange spectrum (Number Theory) (as we briefly described in this post). In particular, since I’m planning to make a series of posts around these works of C. Moreira and J. C. Yoccoz, I think that a discussion of Kaufmann’s proof of Marstrand theorem is the best place to start this project.
–Statement of Marstrand’s theorem–
Define ,
and denote by
the orthogonal projection of the plane
onto the line
. In other words, if we identify
with the real line
(via
, then, given
, we have
.
Theorem 1 (J. Marstrand’s theorem) Let
be a subset with Hausdorff dimension
. Then, its projection
has positive Lebesgue measure for (Lebesgue) almost every
.
An interesting consequence of this theorem (in the context of stable intersections of Cantor sets of the real line) is the following fact:
Corollary 2 Let
and
be two Cantor sets of the real line
such that
Then,
has positive Lebesgue measure for almost every
.
Proof: Firstly, we recall the following well-known property of the Hausdorff dimension: (see K. Falconer’s book for more details). Secondly, we notice that the projection
of
is
where
. Since
for
and we can parametrize
via
with
, we get the desired corollary from Marstrand’s theorem.
Now we turn to the proof of Marstrand’s theorem following the exposition of the book of Palis and Takens (pages 64, 65, 66 and 67).
–Kaufmann’s proof of Marstrand theorem
We begin with a preliminary reduction.
Lemma 3 Let
be a subset of the plane with Hausdorff dimension
. Given
, we can find
such that
and
for all
and
. Here,
denotes the Hausdorff
-measure.
Proof: See theorem 5.6 of K. Falconer’s book.
In particular, during the proof of Marstrand’s theorem, we can assume that verifies
and, for some constant
,
(for any
and
). In fact, given
with
, we can fix
and apply the previous lemma in order to obtain
verifying
and
. Hence, if we can prove Marstrand’s theorem for
(i.e.,
has positive Lebesgue measure for almost every
), the same result holds for
because
.
So, in the sequel, we’ll make the following hypothesis:
Main Assumption. satisfies
,
and
for any
and
.
In this situation, we can define a finite measure on by the formula
. For each
, we denote by
the unique measure on
satisfying
for every continuous function
.
Note that the support of is contained in
. Hence, our task can be reduced to show that the support of
has positive Lebesgue measure for almost every
. In this direction, we have the following simple criterion:
Lemma 4 Let
be a finite measure with compact support on
and denote by
the Fourier transform of
. Suppose that
Then, the support of
has positive Lebesgue measure.
Proof: Plancherel’s theorem guarantees that is a well-defined square-integrable function on
such that
and
. Thus, the support of
(which is the support of
) can’t have zero Lebesgue measure.
In view of this lemma, the proof of Marstrand’s theorem will be complete once we show that for almost every
.
Observe that for all
. Indeed, if this integral vanishes (for some
), then
(where
). In particular,
for almost every
. Since
, we have
, i.e.,
, a contradiction.
Now, we’ll end the proof of theorem 1 by showing that is square-integrable for almost every
. Note that
In particular,
Thus,
Observe that the integral doesn’t depend on the direction of
(but only on
). Therefore, denoting by
, we get
Hence, for any ,
Since we know that the integral of the Bessel function
is convergent, we see that there exists a real number
(independent of
) such that
In particular,
Because
we can concatenate the these two previous estimates to obtain
We claim that the right-hand side of (1) is finite. Indeed, for any , we have
Using the main assumption for any
and
(where
), we obtain
Putting the equations (1) and (2) together, using Fubini’s theorem and letting , we conclude
Therefore, it follows that is square-integrable for almost every
, so that the proof of Marstrand’s theorem is complete.
Dear Matheus: some corrections.
– The statement of Lemma 3 has a typographic error, as the d’-Hausdorff dimension should be smaller than \infty.
– In the paragraph after the Main Assumption, the support of \mu_teta is a subset of \pi_\teta(K).
By: Yuri on May 22, 2009
at 5:52 pm
Dear Yuri: thanks for the corrections!
By: matheuscmss on May 23, 2009
at 10:17 am
Dear Matheus:
For the lemma $4$, if $\eta$ is not absolutely continuous with respect to the Lebesgue measure, whether it is plausible that we write $d\eta (x)=\phi(x) dx$. Whether one should understand it as in the distribution sense.
By: yaoxiao on October 10, 2017
at 1:08 pm
Dear yaoxiao,
You’re right: one should interpret it in the distributional sense.
Best,
Matheus
By: matheuscmss on October 10, 2017
at 1:16 pm