Dimension of projected sets and Fourier restriction

Pdf version here: link.

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 {\mathbb{R}^2}

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 {\mathbb{R}^2} the projections are associated to the points in {\mathbb{S}^1}, namely for {e\in\mathbb{S}^1} one has {\pi_e (x) = (x\cdot e)e}, and so for a given compact set {K} of Hausdorff dimension {0\leq \dim K \leq 1} one asks what can be said about the set of projections for which the dimension is smaller, i.e. {\dim \pi(K) < \dim K}. For {s \leq \dim K}, define the set of directions

\displaystyle E_s (K):= \{e \in \mathbb{S}^1 \,:\, \dim \pi_e (K) < s\}.

We refer to it as to the set of exceptional directions (of parameter {s}). One preliminary result is Marstrand’s theorem:

Theorem 1 (Marstrand) For any compact {K} in {\mathbb{R}^2} s.t. {s<\dim K <1}, one has

\displaystyle |E_s (K)| = 0.

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 {K} compact in {\mathbb{R}^d}, it is {s<\dim K} if and only if there exists a finite positive Borel measure {\mu} supported in {K} such that

\displaystyle I_s(\mu):= \int_{K}{\int_{K}{\frac{d\mu(x)\,d\mu(y)}{|x-y|^s}}}<\infty.

It should be observed that this is also equivalent to the condition on the balls that {\mu(B(x,r))\lesssim r^s} uniformly in {x}. With this lemma it’s a matter of a few lines to prove Marstrand’s theorem: Proof: Take {s < \dim K} and {\mu} a measure given by Frostman’s lemma. We use {\mu} to generate measures {\mu_e} for a.e. direction in {\mathbb{S}^1} such that {I_s (\mu_e)< \infty} for all of them, and thus {\dim \pi_e (K) > s} for a.e. {e}. The measure will be {\mu_e := (\pi_e)_\ast \mu}, i.e. the measure s.t. {\int_{\mathbb{R}}{f}\,d\mu_e = \int{f(x\cdot e)}\,d\mu(x)}. Thus if

\displaystyle \int_{\mathbb{S}^1}{I_s(\mu_e)}\,d\mathcal{H}^1 (e) <\infty,

it will follow that {I_s (\mu_e) < \infty} a.e.. Here {\mathcal{H}^1} is the {1}-dimensional Hausdorff measure. Now by expanding

\displaystyle \int_{\mathbb{S}^1}{I_s(\mu_e)}\,d\mathcal{H}^1 (e) = \int_{\mathbb{S}^1}{\int{\int{\frac{d\mu_e (t)\,d\mu_e (u)}{|t-u|^s}}}}\,d\mathcal{H}^1 (e)

\displaystyle = \int_{\mathbb{S}^1}{\int_{K}{\int_{K}{\frac{d\mu(x)\,d\mu(y)}{|x\cdot e - y\cdot e|^s}}}}\,d\mathcal{H}^1 (e),

and by Fubini

\displaystyle = \int_{K}{\int_{K}{\left(\int_{\mathbb{S}^1}{\frac{d\mathcal{H}^1 (e)}{\left|e\cdot\frac{x-y}{|x-y|}\right|^s}}\right)\frac{d\mu(x)\,d\mu(y)}{|x\cdot e - y\cdot e|^s}}},

and the term in brackets can be easily computed to be {O(1)} since {s<1}. Finiteness of the integral then follows by the choice of {\mu}. \Box

Next question one might ask about {E_s(K)} is what’s its dimension. Coifman proved

Theorem 3 (Coifman)

\displaystyle \dim E_s(K) \leq s.

On the other hand Kaufman and Mattila proved that this result is sharp when {s \approx \dim K}, namely

Theorem 4 (Kaufman & Mattila’s example) For each {0< s <1} there exists a compact set {K} s.t. {\dim K = s} and

\displaystyle \dim \{ e \in\mathbb{S}^1\,:\, \dim \pi_e (K) < \dim K\} = s.

A complementary result was given by Bourgain, who proved that when {s} is sensibly smaller than {\dim K}, Coifman result is no more sharp

Theorem 5 (Bourgain) Under the hypotheses above, one has

\displaystyle \dim E_s (K) \rightarrow 0 \quad \text{ for }\quad s \rightarrow \frac{\dim K}{2}.

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 {P} of points in {\mathbb{R}^2} of cardinality {\# P = n}. For a given {0<s<1}, what is the cardinality of the set of directions {e\in\mathbb{S}^1} s.t.

\displaystyle \# \pi_e (P)\lesssim n^s\quad ?

(the constant in the inequality of course doesn’t depend on {P}). Here the role of Hausdorff dimension is played by the exponent of {n} – the sets {P} are thought of as having dimension {1} then. An answer to this question is the following:

Proposition 6 If {s<1/2} there can be only one direction for which {\# \pi_e (P) \lesssim n^s}.

If {1/2 \leq s < 1} the number of directions is {\lesssim n^{2s-1}}.

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 {\mathcal{L}} be a collection of lines and {P} a collection of points in the plane. Then one can estimate

\displaystyle \#\{(p,\ell)\in P\times \mathcal{L}\quad \text{ s.t. } p \in \ell\} \lesssim (\#P \#\mathcal{L})^{2/3} + \#P + \# \mathcal{L}.

Let’s briefly see the proof of the proposition above: Proof: If {s<1/2}, suppose there are two different such directions, or lines {\ell, \ell'}. Then they provide a set of axes in the plane, and every point in {P} is identified by its coordinates (elements of {\pi_\ell (P)}). But by the hypothesis, there are at most {\#\pi_\ell (P) \# \pi_{\ell'}(P) \lesssim n^{2s}} points in {P}, which is a contradiction since {n^{2s} \ll n}.

As for the second case, let {S} be the set of directions and define the collection of lines {\mathcal{L} := \{\ell = \pi_e^{-1} (\pi_e (P)), e\in S\}}. Then if {\# S = k} (the number we want to estimate) it is exactly {\#\{(p,\ell)\in P\times \mathcal{L}\quad \text{ s.t. } p \in \ell\} = k n} and {\# \mathcal{L} \lesssim k n^s}. Substituting into Szemeredi-Trotter’s inequality,

\displaystyle kn \lesssim (kn^s)^{2/3} n^{2/3} + kn^s + n,

and since {s\geq 1/2} this reduces to the necessary condition {kn \lesssim k^{2/3} n^{2(s+1)/3}}, which is {k \lesssim n^{2s-1}} indeed. \Box

A first remark is that when {s} approaches {1/2} one has {n^{2s-1}} approach {O(\log n)}, which in our dictionary means that the “dimension” of the set of exceptional directions approaches {0}. 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 {L^{8/5} \rightarrow L^4}) for the paraboloid in the finite fields! This is indeed a discrete analogue of the paraboloid restriction problem in {\mathbb{R}^3}, just like the above is a discrete analogue of the Hausdorff dimension of projections problem. A rigorous statement is in [3]. For the sake of completeness:

Proposition 8 Consider the paraboloid {S = \{(\xi, \xi\cdot \xi), \xi \in \mathbb{F}^{2}\},} where {\mathbb{F}} is a finite field in which {-1} is not a square, and equip it with the normalized counting measure {d\sigma}. Define the Fourier extension operator

\displaystyle (g d\sigma)^\vee (x)= \int_{S}{g(\xi)e(x\cdot \xi)}\,d\sigma(\xi),

where {e} is any non-trivial character of {\mathbb{F}}. Then one has

\displaystyle \|(g d\sigma)^\vee \|_{L^{4}(\mathbb{F}^3,dx)} \lesssim \|g\|_{L^{8/5}(S,d\sigma)},

where {dx} is the counting measure on {\mathbb{F}^3}.

(the original proof has a logarithmic loss, namely a factor {\log \# \mathbb{F}} on the RHS, but it can be dealt with – see [2]). The incidence result one exploits here is the inequality

\displaystyle \#\{(p,\ell)\in P\times \mathcal{L}\quad \text{ s.t. } p \in \ell\} \leq \min\{(\# P)^{1/2}\#\mathcal{L} + \# P, (\#\mathcal{L})^{1/2} \#P + \#\mathcal{L}\}

in the plane {\mathbb{F}^2}. Notice that by taking a geometric mean of the terms in the minimum one can get {(\# P \#\mathcal{L})^{3/4} + \#P + \#\mathcal{L}} 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 {\mathbb{R}^2} 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 {\mathbb{R}^3}

Having said quite a lot about the plane case, the next case to be approached is the case of projections in {\mathbb{R}^3}, which have rank 1 or 2, and are described by the grassmannians {\mathcal{G}(3,1)\cong \mathbb{PR}^2} and {\mathcal{G}(3,2)} respectively. For {V \in \mathcal{G}(3,2)} one defines the projection {\pi_V} in the obvious way. In analogy with the previous case, one can prove

Theorem 9 If {K} is a compact set in {\mathbb{R}^3} s.t. {\dim K \leq 2} then

\displaystyle \dim \pi_V (K) = \dim K

for almost every {V\in \mathcal{G}(3,2)}.

Though this is interesting, it is true that “a.e.” in {\mathcal{G}(3,2)} leaves out quite a lot of stuff – there’s plenty of 1-dimensional submanifolds contained in {\mathcal{G}(3,2)}, 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 {\mathcal{G}(3,2)} can be described by the normal vector to the planes, thus by a curve {\gamma(t)} with support on {\mathbb{S}^2}. Some of these families are more general than others: in particular, think of the planes that contain the {z}-axis (in which case {\gamma} is an equator) and compare to those whose normals are described by a general curve with non-vanishing curvature, i.e. for every {t} it is {\mathrm{rank}(\gamma(t),\gamma'(t),\gamma''(t))=3}.

One can suitably modify the proof above to prove Marstrand’s theorem for these restricted families in {\mathbb{R}^3}:

Theorem 10 If {\dim K \leq 1} then

\displaystyle \dim \pi_{V_t}(K) = \dim K \qquad \text{for a.e. }t,

where {V_t = \gamma(t)^{\perp}}.

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, [1]) Assume {\gamma(t)} has non-vanishing curvature; {K} 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} s.t. {\dim_p K > s > 1}, where {\dim_p} is now the packing dimension, there exists {\sigma(s) > 1} such that

\displaystyle \dim_p \pi_{V_t}(K) > \sigma(s) \qquad \text{ for a.e. } t.

It is conjectured that {\sigma(s) = \min\{s,2\}}. 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, [4] ) Consider a restricted family of projections defined by a curve {\gamma} satisfying the curvature condition as before. Suppose {1\leq \dim K \leq 2}. Then

\displaystyle \dim \pi_{V_t} (K) \geq \frac{3\dim K}{4}

for almost every {t}.

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 {I_s (\mu)} of a measure {\mu} can be restated in terms of its Fourier transform only. Indeed, if we denote by {k_s (x)} the tempered distribution that agrees with kernel {|x|^{-s}} away from zero, then

\displaystyle \int{\int{\frac{d\mu(x) d\mu(y)}{|x-y|^s}}} = \int{k_s \ast \mu (x)}\,d\mu(x)

(one can make sense of the integrals at least in a distributional way), and this can be rewritten as

\displaystyle = \int{\widehat{k_s \ast \mu}(\xi)\overline{\widehat{\mu}(\xi)}}\,d\xi = \int{\widehat{k_s}(\xi) \widehat{\mu}(\xi)\overline{\widehat{\mu}(\xi)}}\,d\xi

\displaystyle = c_d \int{\left|\widehat{\mu}(\xi)\right|^2 |\xi|^{s-d}}\,d\xi,

where {d} is the dimension of the space and {c_d |\xi|^{s-d}} is the Fourier transform of {k_s}. This formulation allows us to use tools from harmonic analysis quite directly.

Another thing to notice is that the push-forwarded measure {\mu_e} we defined earlier has Fourier transform that can be written in terms of {\widehat{\mu}}: indeed

\displaystyle \widehat{\mu_e}(t) = \int_{\mathbb{R}}{e^{-2\pi i t s }}\,d\mu_e (s) = \int_{\mathbb{R^d}}{e^{-2\pi i t (x\cdot e)}}\,d\mu(x) = \widehat{\mu}(te).

We can prove a Marstrand theorem in {\mathbb{R}^3} with this formulation of energy.

Theorem 13 (Marstrand’s theorem for lines in {\mathbb{R}^3}) For a.e. {\omega \in \mathbb{S}^2}, {K\subset \mathbb{R}^3} compact with {\dim K \leq 1}, one has

\displaystyle \dim \pi_\omega (K) = \dim K.

Proof: As we did before, we prove that for some {\mu} we have {\int_{\mathbb{S}^2}{I_s (\mu_\omega)}\,d\mathcal{H}^2(\omega) < \infty} for every {s< \dim K}, which implies the result. Fix such an {s} and take by Frostman’s lemma {\mu} s.t. {I_s(\mu)< \infty}. Then

\displaystyle \int_{\mathbb{S}^2}{I_s (\mu_\omega)}\,d\mathcal{H}^2(\omega) = \int_{\mathbb{S}^2}{\int_{\mathbb{R}}{\left|\widehat{\mu_\omega}(t)\right|^2 |t|^{s-1} }\,dt}\,d\mathcal{H}^2(\omega)

\displaystyle =\int_{\mathbb{S}^2}{\int_{\mathbb{R}}{\left|\widehat{\mu}(t\omega)\right|^2 |t|^{s-1} }\,dt}\,d\mathcal{H}^2(\omega) = 2\int_{\mathbb{S}^2}{\int_{0}^{\infty}{\left|\widehat{\mu}(t\omega)\right|^2 |t|^{s-1} t^2 t^{-2} }\,dt}\,d\mathcal{H}^2(\omega),

but this is just integration in polar coordinates, so

\displaystyle = 2\int_{\mathbb{R}^3}{\left|\widehat{\mu}(\xi)\right|^2 |\xi|^{s-3}}\,d\xi < \infty

by the choice of {\mu}. \Box

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 {\gamma} 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 {\gamma(t)}.

cone

So, assume {K} is a compact set in {\mathbb{R}^3} with dimension {1<\dim K <2}. Consider the family of lines through the origin with direction {\gamma(t)} for {\gamma} a circle of constant latitude on the sphere ({t\in \mathbb{S}^1}), say {45^\circ}, and denote by {\pi_t} the orthogonal projection onto {\gamma(t)^{\perp}}. We claim we can prove the following weaker version of the result of Oberlin and Oberlin:

Proposition 14 It holds

\displaystyle \dim \pi_t K \geq \frac{\dim K}{3}

for a.e. {t\in\mathbb{S}^1}.

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 {u<\dim K} and by Frostman lemma you have a measure {\mu} on {K} s.t. {I_u (\mu) < \infty}. We want to prove that {I_s(\mu_t)} is finite for a.e. {t}, where {s < u/3} and {\mu_t} is the push-forward {(\pi_t)_\ast \mu}.

In order to show the finiteness a.e. of {I_s (\mu_t)} we can prove

\displaystyle \int_{\mathbb{S}^1}{I_s(\mu_t)}\,dt<\infty

instead, and this is equivalent to proving

\displaystyle \int_{\mathbb{S}^1}\int_{\mathbb{R}}{|\widehat{\mu_t}(\xi)|^2 |\xi|^{s-1}}\,d\xi \,dt<\infty.

Now we split {\mathbb{R}} into dyadic intervals and estimate equivalently

\displaystyle \sum_{j\in\mathbb{Z}}{2^{j(s-1)}\int_{\mathbb{S}^1}\int_{|\xi|\sim 2^j}{|\widehat{\mu_t}(\xi)|^2}\,d\xi \,dt}. \ \ \ \ \ (1)

 

By what we’ve seen before, {\widehat{\mu_t}(\xi) = \widehat{\mu}(\gamma(t) \xi)}, and thus the integral in the last expression becomes

\displaystyle \int_{\Gamma_j}{|\widehat{\mu}|^2}\,dt \,d\xi,

where {\Gamma_j} is a horizontal slice of cone as follows:

\displaystyle \Gamma_j := \{\xi\gamma(t)\,:\, t\in\mathbb{S}^1, 2^j\leq |\xi|<2^{j+1}\}.

Here, we can actually get rid of this dependence on {j} in the domain by introducing some localized bump function: suppose {\psi} is a Schwartz function s.t. {\widehat{\psi}} is identically {1} on {B(0,2)\backslash B(0,1/2)}, {|\widehat{\psi}|\leq 1} everywhere and has support on {B(0,4) \backslash B(0,1/4)}. Then the expression in (1) is majorized by

\displaystyle \sum_{j\in\mathbb{Z}}{2^{j(s-1)}\int_{\mathbb{S}^1\times \mathbb{R}}{|\widehat{\mu}(\xi\gamma(t)) \widehat{\psi}(2^{-j}\xi \gamma(t))|^2}\,dt\,d\xi}.

Here comes the restriction estimate. It’s a result of Barcelo Taberner (see [5]), and can be stated as follows:

Theorem 15 Let {\Gamma} denote the cone as defined above. Then for {1\leq p < 4/3} we have the a priori estimate

\displaystyle \left\|\widehat{f}\right\|_{L^{p'/3} (\Gamma, d\sigma)} \lesssim_{p} \|f\|_{L^p (\mathbb{R}^3)},

where {\frac{1}{p}+\frac{1}{p'}=1} and {d\sigma} is the measure that in polar coordinates is {d\theta dr} – i.e. {r^{-1} dS}, where {r} is the distance from the cone vertex and {dS} 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 {d\sigma} is exactly {dt \, d\xi}, so we can apply the theorem straight away. We have

\displaystyle \int_{\Gamma}{|\widehat{\mu} \widehat{\psi_{2^{-j}}}|^2}\,dt\,d\xi = \left\|\widehat{\mu \ast \psi_{2^{-j}}}\right\|_{L^2(\Gamma,d\sigma)}^2 \lesssim \|\mu\ast \psi_{2^{-j}}\|_{L^{6/5}(\mathbb{R}^3)}^2,

where {f_\lambda (x) := \lambda^{-d} f(\lambda^{-1} x)}; it’s too hard to estimate the {6/5} norm directly so we use logarithmic interpolation instead:

\displaystyle \|\mu\ast \psi_{2^{-j}}\|_{L^{6/5}(\mathbb{R}^3)}^2 \leq \|\mu\ast \psi_{2^{-j}}\|_{L^{1}(\mathbb{R}^3)}^{4/3} \|\mu\ast \psi_{2^{-j}}\|_{L^{2}(\mathbb{R}^3)}^{2/3}.

By assumption {\mu} has finite mass, and therefore the {L^1} norm above is {O(1)}. We also have by assumption that

\displaystyle \sum_{j\in\mathbb{Z}}{2^{j(u-3)}\left\|\widehat{\mu\ast\psi_{2^{-j}}}\right\|_{L^2(\mathbb{R}^3)}^2} < C < \infty,

so that for {j\geq 0} we have

\displaystyle \sum_{j\geq 0}{2^{j(s-1)}\|\mu\ast \psi_{2^{-j}}\|_{L^{2}(\mathbb{R}^3)}^{2/3}} \leq \left(\sum_{j\geq 0}{2^{j(u-3)}\|\mu\ast \psi_{2^{-j}}\|_{L^{2}(\mathbb{R}^3)}^{2}}\right)^{1/3}\left(\sum_{j\geq 0}{2^{\frac{3}{2}j(s-1-\frac{u}{3}+1)}}\right)^{2/3},

in which the first factor is finite by the choice of {\mu} and the second one is finite iff {s-\frac{u}{3} <0 }, i.e. for all {s < \frac{u}{3}}, which is what we want.

As for the remaining {j<0} part of the sum, we estimate simply by Young inequality

\displaystyle \sum_{j<0}{2^{j(s-1)}\|\mu\ast\psi_{2^{-j}}\|_{L^2}^{2/3}}\leq\sum_{j<0}{2^{j(s-1)}\|\mu\|_{L^1}^{2/3}\|\psi_{2^{-j}}\|_{L^2}^{2/3}}

\displaystyle \lesssim \sum_{j<0}{2^{j(s-1)}(2^{3j/2})^{2/3}\|\psi\|_{L^2}^{2/3}}\lesssim \sum_{j<0}{2^{js}} < \infty,

because {s} is positive. Thus we’ve proved {I_s (\mu_t)} is a.e. finite for all {s < u/3}, and by taking the supremum {\dim \pi_t K \geq \frac{\dim K}{3}} for a.e. {t}. \Box

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 {1/3} because of the weird exponent {6/5} 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.

 

References:

[1] K. Fässler, T. Orponen, On restricted families of projections in {\mathbb{R}^3}, arXiv:1302.6550 [math.CA].

[2] A. Lewko, M. Lewko, Endpoint restriction estimates for the paraboloid over finite fields, arXiv:1009.3080 [math.CA].

[3] G. Mockenhaupt, T. Tao, Restriction and Kakeya phenomena for finite fields, Duke Math. J. Volume 121, Number 1 (2004), 35-74.

[4] D. Oberlin, R. Oberlin, Application of a Fourier restriction theorem to certain families of projections in {\mathbb{R}^3}, arXiv:1307.5039 [math.CA].

[5] B. Barcelo Taberner, On the restriction of the Fourier transform to a conical surface, Trans. Amer. Math. Soc. 292 (1985), 321-333.

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