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I had a nice discussion with Tuomas after the very nice analysis seminar he gave for the harmonic analysis working group a while ago – he talked about the behaviour of Hausdorff dimension under projection operators and later we discussed the connection with Fourier restriction theory. Turns out there are points of contact but the results one gets are partial, and there are some a priori obstacles.
What follows is an account of the discussion. I will summarize his talk first.
1. Summary of the talk
1.1. Projections in
The problem of interest here is to determine whether there is any drop in the Hausdorff dimension of fractal sets when you project them on a lower dimensional vector space, and if so what can be said about the set of these “bad” projections. This is a very hard problem in general, so one has to start with low dimensions first. In the projections are associated to the points in , namely for one has , and so for a given compact set of Hausdorff dimension one asks what can be said about the set of projections for which the dimension is smaller, i.e. . For , define the set of directions
We refer to it as to the set of exceptional directions (of parameter ). One preliminary result is Marstrand’s theorem:
Theorem 1 (Marstrand) For any compact in s.t. , one has
In other words, the dimension is conserved for a.e. direction. The proof of the theorem relies on a characterization of dimension in terms of energy:
Theorem 2 (Frostman’s lemma) For compact in , it is if and only if there exists a finite positive Borel measure supported in such that
It should be observed that this is also equivalent to the condition on the balls that uniformly in . With this lemma it’s a matter of a few lines to prove Marstrand’s theorem: Proof: Take and a measure given by Frostman's lemma. We use to generate measures for a.e. direction in such that for a.e. . The measure will be , i.e. the measure s.t. . Thus if
it will follow that a.e.. Here is the -dimensional Hausdorff measure. Now by expanding
and by Fubini
and the term in brackets can be easily computed to be since . Finiteness of the integral then follows by the choice of .
Next question one might ask about is what's its dimension. Coifman proved
Theorem 3 (Coifman)
On the other hand Kaufman and Mattila proved that this result is sharp when , namely
Theorem 4 (Kaufman & Mattila’s example) For each there exists a compact set s.t. and
A complementary result was given by Bourgain, who proved that when is sensibly smaller than , Coifman result is no more sharp
Theorem 5 (Bourgain) Under the hypotheses above, one has
An interesting fact is that these problems have combinatorial analogues in discrete settings, which display similar properties. One example, related to this last theorem, is the following: Consider a collection of points in of cardinality . For a given , what is the cardinality of the set of directions s.t.
(the constant in the inequality of course doesn’t depend on ). Here the role of Hausdorff dimension is played by the exponent of – the sets are thought of as having dimension then. An answer to this question is the following:
Proposition 6 If there can be only one direction for which .
If the number of directions is .
The proof of this statement relies (for its non trivial part) on a theorem of incidence geometry of Szemeredi and Trotter, namely
Theorem 7 (Szemeredi-Trotter) Let be a collection of lines and a collection of points in the plane. Then one can estimate
Let’s briefly see the proof of the proposition above: Proof: If , suppose there are two different such directions, or lines . Then they provide a set of axes in the plane, and every point in is identified by its coordinates (elements of ). But by the hypothesis, there are at most points in , which is a contradiction since .
As for the second case, let be the set of directions and define the collection of lines . Then if (the number we want to estimate) it is exactly and . Substituting into Szemeredi-Trotter's inequality,
and since this reduces to the necessary condition , which is indeed.
A first remark is that when approaches one has approach , which in our dictionary means that the “dimension” of the set of exceptional directions approaches . This is the exact translation of Bourgain’s result in this discrete setting.
A second remark is that at this point we can already see one of the points of contact with Fourier restriction problems. They are indeed connected by incidence geometry: using a Szemeredi-Trotter-like lemma one can prove a restriction estimate (namely ) for the paraboloid in the finite fields! This is indeed a discrete analogue of the paraboloid restriction problem in , just like the above is a discrete analogue of the Hausdorff dimension of projections problem. A rigorous statement is in . For the sake of completeness:
Proposition 8 Consider the paraboloid where is a finite field in which is not a square, and equip it with the normalized counting measure . Define the Fourier extension operator
where is any non-trivial character of . Then one has
where is the counting measure on .
(the original proof has a logarithmic loss, namely a factor on the RHS, but it can be dealt with – see ). The incidence result one exploits here is the inequality
in the plane . Notice that by taking a geometric mean of the terms in the minimum one can get at the RHS, which has a worse exponent than Szemeredi-Trotter above. Indeed, one can prove that in the finite field setting this is optimal, and thus any proof of the Szemeredi-Trotter inequality in the real case can’t rely on algebraic means only – it needs to incorporate some topology or as well. This point is made very clear in the following post by Tao, in which he proves Szemeredi-Trotter via the crossing number inequality: http://terrytao.wordpress.com/2007/09/18/the-crossing-number-inequality/
A proof of the proposition above would make this post even too long, and hence we refer the reader to the papers cited above.
As passing to discrete settings is thought to help highlighting the essence of a problem, one could expect both the original problems to be instances of some finer geometric incidence theory. A satisfactory incidence theory of tubes is indeed the current Holy Grail of harmonic analysis I think (see e.g. the Kakeya conjecture(s)), and if I understood Tuomas correctly, incidence theory of tubes could prove very powerful in handling the higher dimensional cases of the Hausdorff dimension problems I’ve described here so far.
1.2. Projections in
Having said quite a lot about the plane case, the next case to be approached is the case of projections in , which have rank 1 or 2, and are described by the grassmannians and respectively. For one defines the projection in the obvious way. In analogy with the previous case, one can prove
Theorem 9 If is a compact set in s.t. then
for almost every .
Though this is interesting, it is true that “a.e.” in leaves out quite a lot of stuff – there’s plenty of 1-dimensional submanifolds contained in , for example. It is then natural to ask whether one can get some results for these particular restricted families of projections. A 1-dimensional submanifold of can be described by the normal vector to the planes, thus by a curve with support on . Some of these families are more general than others: in particular, think of the planes that contain the -axis (in which case is an equator) and compare to those whose normals are described by a general curve with non-vanishing curvature, i.e. for every it is .
One can suitably modify the proof above to prove Marstrand’s theorem for these restricted families in :
Theorem 10 If then
For bigger dimension something can be said too in the case of restricted families of projections (and this is a result of Tuomas himself), and here the curvature condition is fundamental
Theorem 11 (K. Fässler, T. Orponen, ) Assume has non-vanishing curvature; is an analytic set, i.e. the continuous image of a Polish space (a separable metric space; Borel sets are analytic, for example). For every s.t. , where is now the packing dimension, there exists such that
It is conjectured that . I should also mention that this has been partially extended to cover the Hausdorff dimension case as well. One case in which this happened allows me to finally get to the next section, about Fourier restriction and its relationship to this subject.
2. Applications of Fourier restriction
The result I’m talking about is the following
Theorem 12 (D. Oberlin, R. Oberlin,  ) Consider a restricted family of projections defined by a curve satisfying the curvature condition as before. Suppose . Then
for almost every .
This is somewhat complementary to theorem 11. What’s interesting to me is that it is proved by using results in Fourier restriction theory. I will now start to make the way into the matter.
First thing to notice is that the energy of a measure can be restated in terms of its Fourier transform only. Indeed, if we denote by the tempered distribution that agrees with kernel away from zero, then
(one can make sense of the integrals at least in a distributional way), and this can be rewritten as
where is the dimension of the space and is the Fourier transform of . This formulation allows us to use tools from harmonic analysis quite directly.
Another thing to notice is that the push-forwarded measure we defined earlier has Fourier transform that can be written in terms of : indeed
We can prove a Marstrand theorem in with this formulation of energy.
Theorem 13 (Marstrand’s theorem for lines in ) For a.e. , compact with , one has
Proof: As we did before, we prove that for some we have for every , which implies the result. Fix such an and take by Frostman's lemma s.t. . Then
but this is just integration in polar coordinates, so
by the choice of .
This can be obviously generalized to lines in any dimension.
As stated in the previous section, it is a relevant question whether you can say something relevant for restricted families of projections. Consider then the case of a curve which is just a circle of constant latitude on the sphere. You might think of it as given by the intersection of a cone with the sphere (wink, wink), and then ask if you have some good lower bound for the drop in dimension when projecting onto directions .
So, assume is a compact set in with dimension . Consider the family of lines through the origin with direction for a circle of constant latitude on the sphere (), say , and denote by the orthogonal projection onto . We claim we can prove the following weaker version of the result of Oberlin and Oberlin:
Proposition 14 It holds
for a.e. .
The interesting thing is that we can prove it by means of a Fourier restriction estimate! Let me show how: Proof: We proceed as in the proof of Marstrand theorem. Choose and by Frostman lemma you have a measure on s.t. . We want to prove that is finite for a.e. , where and is the push-forward .
In order to show the finiteness a.e. of we can prove
instead, and this is equivalent to proving
where is a horizontal slice of cone as follows:
Here, we can actually get rid of this dependence on in the domain by introducing some localized bump function: suppose is a Schwartz function s.t. is identically on , everywhere and has support on . Then the expression in (1) is majorized by
Here comes the restriction estimate. It’s a result of Barcelo Taberner (see ), and can be stated as follows:
Theorem 15 Let denote the cone as defined above. Then for we have the a priori estimate
where and is the measure that in polar coordinates is – i.e. , where is the distance from the cone vertex and is the surface measure induced by the Lebesgue measure.
The necessity of such a choice for the measure is seen by a scaling argument as usual. Notice in our notation is exactly , so we can apply the theorem straight away. We have
where ; it’s too hard to estimate the norm directly so we use logarithmic interpolation instead:
By assumption has finite mass, and therefore the norm above is . We also have by assumption that
so that for we have
in which the first factor is finite by the choice of and the second one is finite iff , i.e. for all , which is what we want.
As for the remaining part of the sum, we estimate simply by Young inequality
because is positive. Thus we’ve proved is a.e. finite for all , and by taking the supremum for a.e. .
I find it pretty amusing that Fourier restriction estimates can have such applications to geometric problems (and not just PDEs). Nevertheless, we don't get the full result (not even in the Oberlin paper) that we'd expect – i.e. the conjecture that the dimension doesn't change. Notice that we get the factor of because of the weird exponent we get from the restriction estimate for the cone, but this exponent is necessary as mentioned above, and thus it seems we can't squeeze any more juice from this method. This is a little frustrating, essentially because I can't make much sense of it. Maybe I have too high expectations, but I can't figure out what exactly stands in the way. As mentioned in the introduction, Tuomas showed me a possible obstacle in this sense, but I have to think about it for a while more. I'll come back to this point later on.
 K. Fässler, T. Orponen, On restricted families of projections in , arXiv:1302.6550 [math.CA].
 A. Lewko, M. Lewko, Endpoint restriction estimates for the paraboloid over finite fields, arXiv:1009.3080 [math.CA].
 G. Mockenhaupt, T. Tao, Restriction and Kakeya phenomena for finite fields, Duke Math. J. Volume 121, Number 1 (2004), 35-74.
 D. Oberlin, R. Oberlin, Application of a Fourier restriction theorem to certain families of projections in , arXiv:1307.5039 [math.CA].
 B. Barcelo Taberner, On the restriction of the Fourier transform to a conical surface, Trans. Amer. Math. Soc. 292 (1985), 321-333.