1 Introduction and overview of paper↩︎

Let \(\mu\) be a finite compactly supported Borel measure on \(\mathbb{R}^d\). The Fourier restriction or extension problem asks when it is meaningful to extend the Fourier transform of an arbitrary \(L^{p}(\mu)\) function to a function in \(L^{q}(\mathbb{R}^d)\). This is certainly not always possible and the values of \(p\) and \(q\) for which such an extension does hold depend on \(\mu\) in a subtle way. See Section 2.2 for a more detailed discussion of this problem. A different but related problem is the \(L^p\)-improving problem, which is to determine when convolution with \(\mu\) improves the integrability of all \(L^p\) functions. See Section 2.3 for a more detailed discussion of this problem. Yet another distinct but related problem is the Fourier decay problem which asks how fast the Fourier transform of \(\mu\) decays. This can be done in an \(L^\infty\) sense, leading to the Fourier dimension which captures the optimal uniform decay rate, or in a suitable average sense which leads to the Fourier spectrum, see Section 2.4.1 for the definitions. The precise values of the Fourier dimension and spectrum also depend on \(\mu\) in a subtle way.

The Fourier restriction, \(L^p\)-improving, and Fourier decay problems are connected by the fact that they are all inhibited by ‘flatness’. That is, if \(\mu\) puts a lot of mass in a relatively flat part of space, then one would expect the available \(L^p\)–\(L^q\) extension, \(L^p\)-improving, and Fourier decay estimates to be weaker. In the restriction literature, ‘flatness’ has often been captured by Knapp style examples. Our first goal is to formulate a general Knapp example and to precisely describe its consequences for general restriction/extension estimates, see Theorem 3. This follows the classical Knapp style argument but serves to unify various estimates from the literature. Our next goal is to establish a Knapp style obstruction in the \(L^p\)-improving problem which extends and simplifies previous results, see Theorem 5. Turning to Fourier decay, we next aim to prove a novel Knapp style obstruction in the Fourier decay problem, that is, to bound the Fourier dimension or spectrum from above in terms of ‘quantitative flatness’. This turns out to be quite subtle and we spend some time formulating various different approaches to this (see Corollary 8, and Theorems 12, 13, and 14) and discussing why a natural conjecture in this direction fails (see, in particular, Example 5). Next we precisely characterise \(L^2\)-flattening in terms of the Fourier spectrum (see Theorem 16) and this allows us to show that the strongest form of quantitative flattening we consider also prohibits \(L^2\)-flattening. It is worth noting here that the two distinct uses of ‘flattening’ in the previous sentence refer to very different things and are in fact going in opposite directions: if a measure is too ‘flat’, then its Fourier transform is not good at ‘flattening’ other measures!

Once our abstract obstructions to Fourier restriction, \(L^p\)-improving, Fourier decay, and \(L^2\)-flattening are established, our goal shifts to applications of these results. We spend the rest of the paper establishing quantitative flatness in various situations. We start with some basic estimates, recovering a result of Mitsis which relates the Fourier dimension and Frostman dimension and a basic obstruction to restriction estimates due to Hambrook–Łaba (see Theorem 18). Next we consider the classical setting for Fourier restriction, that is, when \(\mu\) is the surface measure on a \(C^2\) surface. Here there is currently considerable interest in Fourier dimension estimates where it has recently been proved that the Fourier dimension of a smooth hypersurface of constant rank is equal to its rank, see papers of Zhu [1], [2], and also [3]. We are able to consider arbitrary co-dimension (that is, beyond co-dimension 1) and also variable rank. We prove that the Fourier dimension of the surface measure on a \(C^2\) surface is bounded above by the smallest ambient rank of a point on the surface (see Theorem 20). We also prove that the Fourier dimension of a \(C^2\) surface itself is bounded above by the largest ambient rank of a point on the surface, provided this number is sufficiently small (see Theorem 21). Both of these estimates are best possible and resolve, or make progress, on a folklore conjecture about how the Fourier dimension depends on the rank of a surface. In a complementary direction, we prove that if a submanifold of \(\mathbb{R}^d\) has sufficiently small dimension relative to the ambient spatial dimension \(d\), then it cannot be Salem and we give a bound for the Fourier dimension which goes to zero as \(d \to \infty\) (see Corollaries 23, 24 and 25). For example, we prove that the Fourier dimension of a smooth curve in \(\mathbb{R}^d\) is at most \(4/(d+1)\) and the Fourier dimension of the arclength measure is at most \(4/(d+2)\).

The general nature of our flatness framework means it is amenable to various geometric observations which a priori have nothing to do with Fourier decay, such as tube nullity (see Theorem 30), the Assouad spectrum of slices (see Theorem 31, and Theorem 34 for nonlinear slices) and the Assouad spectrum of projections (see Theorem 36).

In the final part of the paper, we consider explicit examples coming from the fractal geometry literature, including self-affine sets (see Theorem 39), limit sets of Kleinian groups (see Theorem 43), and Julia sets of rational maps (see Theorem 42). We use geometric features of these examples to establish quantitative flatness which in turn yields obstructions to the Fourier restriction and decay problems. More precisely, we use the non-conformality in the construction of self-affine sets to force flatness inside basic construction cylinders and, in the case of Kleinian limit sets and Julia sets, we use parabolicity to force flatness in a (flat) neighbourhood of the parabolic fixed points. Estimating the Fourier dimension in these three classes is notoriously difficult (and generally open) and often any positive results are non-quantitative and give rise to mysterious numerical values, see [4]–[9]. By contrast, the bounds we obtain are clean and depend only on familiar parameters such as the entropy, Lyapunov exponents, parabolic ranks, and petal numbers. The bounds we obtain often allow us to rule out the possibility that the measures in question are Salem. Moreover, we are able to exhibit examples in each class where the Fourier dimension is arbitrarily small while the Hausdorff dimension remains bounded away from zero. A far as we are aware, we provide the first non-trivial negative results for the Fourier restriction, \(L^p\)-improving, or Fourier decay problems for these well-studied classes of fractal.

We refer the reader to [10] for a comprehensive treatment of Fourier analysis and its applications in fractal geometry, as well as the Fourier restriction and decay problems. For more background on Fourier analysis in general, see [11]–[13], and for fractal geometry, see [14]–[16].

2 Quantitative flatness: abstract obstructions in Fourier analysis↩︎

2.1 Key definitions and terminology↩︎

Throughout the paper we consider non-zero finite compactly supported Borel measures \(\mu\) on \(\mathbb{R}^d\) and we are interested in their Fourier analytic properties. Our aim is to use quantified information pertaining to how ‘flat’ or ‘concentrated’ the measure is to obstruct or inhibit what might be perceived as ‘good’ Fourier analytic behaviour. In order to do this we introduce \(\gamma\)-flat measures, which use a parameter \(\gamma\) to ensure that a disproportionately large amount of the measure is concentrated inside a (usually anisotropic) cuboid. We say \(Q \subseteq \mathbb{R}^d\) is a cuboid if \[Q=g\left(I_1 \times \cdots \times I_d \right)+z\] where \(g \in SO(d)\) is an orthogonal transformation of \(\mathbb{R}^d\), \(z \in \mathbb{R}^d\) is a translate, and \(I_j\) are non-empty non-degenerate compact intervals in \(\mathbb{R}\) centred at the origin. We refer to \(z\) as the centre of the cuboid \(Q\).

Here and throughout, we use the notation \(A\lesssim B\) to mean that there exists a uniform constant \(c\) such that \(A\leqslant cB\). The constant \(c\) is allowed to depend on the ambient spatial dimension \(d\) and on the measure in question. It can also depend on parameters which are fixed, e.g. the flatness parameter \(\gamma\). But it cannot depend on variables, such as an element in the sequence of cuboids used to witness flatness or on particular length scales used in the relevant proof. We also write \(A\approx B\) if \(A\lesssim B\) and \(B\lesssim A\). We write \(\mathcal{L}^d\) for the \(d\)-dimensional Lebesgue measure on \(\mathbb{R}^d\) and it is useful to keep in mind the simple fact that \[\mathcal{L}^d(Q) = \prod_{j=1}^d |I_j|\] where \(|I_j|>0\) is the length of the interval \(I_j\).

We now introduce the first of two key definitions which will be used throughout the paper.

If the cuboids are isotropic (that is, they are in fact cubes), then being \(\gamma\)-flat does not really imply anything about flatness, but rather that the measure is concentrated in a small ball. This information is precisely described by the Frostman and box dimensions of the measure, see Section 3 for precise definitions. However, the isotropic case is the trivial case where we will find the least interesting consequences. For this reason we chose to use the term ‘flat’ rather than, for example, ‘concentrated’ to reflect that the anisotropic case, which does imply ‘flatness’, is the most important.

It is worth keeping in mind that \(\gamma\)-flatness gets harder to satisfy as \(\gamma\) decreases. Indeed, an easy pigeonholing argument shows that all measures are \(1\)-flat and there do exist \(0\)-flat measures, as we will see below. Therefore, the interesting range is \(\gamma \in [0,1)\).

Often we will need a refinement of the notion of \(\gamma\)-flatness where we not only pay attention to the mass of the cuboids, but also how the Lebesgue volume of the cuboid relates to the shortest sidelength. In some sense this is also quantifying ‘how flat’ the cuboids are. This is captured by the following, which is our second key definition.

In the above, \(\upsilon = d\) corresponds to the least interesting isotropic case where the \(Q_k\) are (up to constants) cubes of volume \(\delta_k^d\). On the other hand, \(\upsilon=1\) applies to situations in the other extreme where the cuboids have sidelengths (up to constants) \(\delta_k \times 1 \times \cdots \times 1\).

To the best of our knowledge, the concept of being \(\gamma\)-flat (Definition 1) is new in this level of generality, although it has appeared before in the context of measures on submanifolds. In particular, the Oberlin condition from [17] (see also [18]) is essentially the negation of being \(\gamma\)-flat. This consideration also leads to Oberlin’s affine dimension, introduced in [19], which involves constructing an affine Hausdorff measure based on covers by cuboids. Moreover, these quantified descriptions of flatness—along with the classical Knapp examples—are known to be related to Fourier restriction, see [10], [17]–[20] and the references therein. The concept of being \((\gamma,\upsilon)\)-flat (Definition 2) appears to be genuinely new, although it is so simple and natural that it may very well have been considered elsewhere.

In the following subsections, we will deduce consequences of \(\gamma\)-flatness and \((\gamma,\upsilon)\)-flatness concerning several well-known problems in Fourier analysis.

2.2 Fourier restriction: a general Knapp example↩︎

The Fourier restriction problem asks when it is meaningful to restrict the Fourier transform of a function to a set of measure zero. Particular instances of this problem have become some of the most famous and long-standing problems in harmonic analysis, going back to work by Stein (see [21], [22]), and have deep connections with geometric measure theory and PDEs. Some of the most important examples include smooth co-dimension 1 manifolds such as the sphere, cone and hyperboloid. Here and throughout, we write \(p'\) and \(q'\) for the Hölder conjugates of \(p,q \in [1,\infty]\), defined by \(\frac{1}{p} + \frac{1}{p'} = \frac{1}{q} + \frac{1}{q'}=1\).

More precisely, the Fourier restriction problem asks, given a measure \(\mu\) supported on a Lebesgue null set in \(\mathbb{R}^d\), for which values of \(q'\in[1,2)\) and \(p'\in[1,\infty]\) is the Fourier transform \(L^{q'}(\mathbb{R}^d) \to L^{p'}(\mu)\) bounded. That is, we say \(\mu\) satisfies the \(R(q'\to p')\) restriction estimate if \[\label{eq:restriction} \| \widehat{f}\, \|_{L^{p'}(\mu)} \lesssim \| f \|_{{L^{q'}(\mathbb{R}^d)}}\tag{1}\] holds for all \(f\in L^{q'}(\mathbb{R}^d)\). By \(L^p\) duality, the \(R(q'\to p')\) restriction estimate 1 is equivalent to the \(E(p\to q)\) extension estimate \[\label{eq:extension} \| \widehat{f\mu} \|_{L^{q}(\mathbb{R}^d)} \lesssim \| f \|_{L^{p}(\mu)}\tag{2}\] holding for all \(f\in L^{p}(\mu)\). Note that we restrict our attention to \(q>2\) because if \(q=q'=2\) then we consider \(f \in L^2(\mathbb{R}^d)\) and the Fourier transform of such functions is only defined almost everywhere and so we cannot make sense of the restriction to a Lebesgue null set. We will refer to the extension estimates throughout the paper, noting the above equivalence with restriction. Our goal will be to impose restrictions on the pairs \((p,q)\) for which a given measure satisfies the \(E(p\to q)\) extension estimate 2 .

The classical ‘Knapp example’ in restriction theory establishes a necessary condition for restriction estimates to hold for the surface measure on the sphere. The idea is to choose a small ‘cap’ which contains a relatively large amount of measure in a relatively flat piece of the surface in question. Recall that genuinely flat sets have no Fourier decay, and therefore no Fourier restriction estimates are possible for them. Knapp examples extend this idea by showing that ‘roughly flat’ pieces provide obstructions to obtaining good restriction and extension estimates.

First we establish an abstract Knapp obstruction which we will then apply to several examples. This construction is a modification of the standard Knapp example but we are unaware of it appearing in the literature in this generality. The most general versions we are aware of are [20] and [18] (see [17]) but these are still in the context of measures on smooth surfaces.

An immediate corollary of the above and the observation that all measures are 1-flat is the following universal obstruction.

The obstruction obtained in Theorem 3 gets stronger as \(\gamma\) decreases. When \(\gamma=1\) it does not provide any non-trivial information (since all measures are 1-flat) and when \(\gamma=0\) it provides the strongest possible conclusion which is that there are no non-trivial restriction or extension estimates. In particular, it forces \(q'=1\) and the Fourier transform of an \(L^1\) function is continuous and so in \(L^{p'}(\mu)\) for all \(p'\), recalling that \(\mu\) is compactly supported. Therefore, 0-flatness forbids all but the trivial estimates which hold for all measures.

As a warm up to more complicated examples, we show how Theorem 3 applies in some classical settings. These examples are, of course, not new.

2.3 \(L^p\)-improving estimates↩︎

Another important problem in harmonic analysis is to determine certain \(L^p\)-improving properties for the convolution operator, see [18], [23]–[25]. More precisely, we say \(\mu\) satisfies the \(I(p \to q)\) improving estimate if \[\label{improvingest} \|f \ast \mu \|_{L^q(\mathbb{R}^d)} \lesssim \|f \|_{L^p(\mathbb{R}^d)}\tag{3}\] holds for all \(f \in L^p(\mathbb{R}^d)\). That is, \(\mu\) ‘improves’ all \(L^p\) functions to \(L^q\) functions when used to define a convolution operator. Stein [24] initiated the study of such \(I(p \to q)\) improving estimates and proposed the problem of trying to characterise which \(I(p \to q)\) improving estimates hold in terms of the ‘size’ of \(\mu\). It is straightforward to see that all finite measures satisfy the \(I(p \to p)\) improving estimate and so the challenge is to establish 3 for \(q>p\).

The fact that being \(\gamma\)-flat inhibits the \(L^p\)-improving properties of a measure \(\mu\) appears in work of Oberlin [18] where certain measures on hypersurfaces were considered, see also [17]. Oberlin also connected the problem to affine dimension (of sets) in [19], which is itself directly connected to \(\gamma\)-flatness. In particular, Oberlin obtains the same numerology as we do below, see [18], and also [19]. As with Theorem 3, we are not aware of the following result appearing in the literature in this generality, but it is a generalisation of a well-known phenomenon.

As with Corollary 4, an immediate corollary of the above and the observation that all measures are 1-flat is the following universal obstruction.

Further, when \(\mu\) satisfies the strongest form of flatness, then we see that no non-trivial \(L^p\)-improving estimates are possible.

2.4 Fourier decay: bounding the Fourier dimension and spectrum↩︎

2.4.1 The Fourier dimension and Fourier spectrum↩︎

In this section we recall the definitions of the Fourier dimension and spectrum and describe some of their basic properties. Decay of the Fourier transform is a fundamental property in Fourier analysis and describes many geometric and analytic features of the measure in question. The optimal decay rate is described by the Fourier dimension of \(\mu\), defined by \[\dim_{\mathrm{F}}\mu = \sup\Big\{ s \in \mathbb{R}: \sup_{\xi\in\mathbb{R}^d}\big| \widehat{\mu}(\xi) \big|^2|\xi|^{s}<\infty \Big\}.\] The Fourier dimension of a set \(X \subseteq \mathbb{R}^d\) is then \[\dim_{\mathrm{F}}X = \sup\Big\{ s\in[0,d] : \exists \mu\text{ on }X \text{ such that } \sup_{\xi\in\mathbb{R}^d}\big| \widehat{\mu}(\xi) \big|^2|\xi|^{s}<\infty \Big\}.\] The Hausdorff dimension of \(X\), denoted by \(\dim_{\mathrm{H}}X\), may also be characterised by Fourier decay, but in an \(L^2\) sense. This correspondence opens the door to many deep connections between fractal geometry and Fourier analysis. In particular, \[\dim_{\mathrm{H}}X = \sup\bigg\{ s\in[0,d] :\exists \mu\text{ on }X \text{ such that } \int_{\mathbb{R}^d} \big| \widehat{\mu}(\xi) \big|^{2}|\xi|^{s - d} \,d\xi < \infty \bigg\}.\] The analogue for measures is the Sobolev dimension of \(\mu\) which is defined by \[\dim_{\mathrm{S}}\mu = \sup\bigg\{ s \in \mathbb{R} : \int_{\mathbb{R}^d} \big| \widehat{\mu}(\xi) \big|^{2}|\xi|^{s - d} \,d\xi < \infty \bigg\}.\] When it is less than \(d\) (which is typical for us), the Sobolev dimension of a measure is also known as the energy dimension, the correlation dimension, or the \(L^2\)-dimension. We say a set \(X\) is Salem if \(\dim_{\mathrm{H}}X = \dim_{\mathrm{F}}X\). Such sets have ‘optimal Fourier analytic structure’ and are useful in many applications. As such, it is desirable to determine whether specific examples are Salem or not. Similarly, we say a measure \(\mu\) on \(\mathbb{R}^d\) is Salem if \(\dim_{\mathrm{H}}\mu = \min\{\dim_{\mathrm{F}}\mu,d\}\). If \(\mu\) is supported on a Lebesgue null set, as most of the measures we consider are, then \(\dim_{\mathrm{F}}\mu \leqslant d\) and the definition of a Salem measure simplifies. The Hausdorff dimension of \(\mu\) is defined by \[\dim_{\mathrm{H}}\mu = \inf\{ \dim_{\mathrm{H}}E: \mu(E)>0\}.\] Given that the Fourier dimension is about \(L^\infty\) control of the Fourier transform and the Hausdorff and Sobolev dimensions are about \(L^2\) control, it is natural to explore \(L^p\) control for intermediate \(p\) in an appropriate sense. This is captured by the Fourier spectrum, originally defined in [26], which is a family of dimensions that continuously interpolate between the Fourier and Hausdorff dimensions for sets and the Fourier and Sobolev dimensions for measures. First, define the \((s,\theta)\)-energies for \(s \in \mathbb{R}\) and \(\theta\in(0,1]\) as \[\mathcal{J}_{s,\theta} (\mu) = \bigg( \int_{\mathbb{R}^d} \big| \widehat{\mu}(\xi) \big|^{\frac{2}{\theta}} |\xi|^{\frac{s}{\theta}-d} \,d\xi \bigg)^\theta,\] and for \(\theta = 0\) as \[\mathcal{J}_{s,0}(\mu) = \sup_{\xi\in\mathbb{R}^d}\big| \widehat{\mu}(\xi) \big|^2|\xi|^{s}.\] Then the Fourier spectrum of \(\mu\) at \(\theta\in[0,1]\) is given by \[\dim^\theta_{\mathrm{F}}\mu = \sup\{ s \in \mathbb{R} : \mathcal{J}_{s,\theta}(\mu)<\infty \},\] and the Fourier spectrum of \(X\) at \(\theta\in[0,1]\) is \[\dim^\theta_{\mathrm{F}}X = \sup\{ s\in[0,d] :\exists \mu\text{ on }X \text{ such that } \mathcal{J}_{s,\theta}(\mu) < \infty \}.\] We may always assume the \(\mu\) used to witness the Fourier dimension and spectrum of \(X\) are compactly supported, even when \(X\) is unbounded, see [27] and [26]. The dimensions mentioned above satisfy \[0 \leqslant\dim_{\mathrm{F}}X = \dim_{\mathrm{F}}^0 X \leqslant\dim^\theta_{\mathrm{F}}X\leqslant\dim_{\mathrm{F}}^1 X = \dim_{\mathrm{H}}X \leqslant d.\] Moreover, the Fourier spectrum \(\dim^\theta_{\mathrm{F}}X\) is non-decreasing and continuous for \(\theta\in[0,1]\). For measures, the Fourier spectrum is concave, continuous on \((0,1]\) and satisfies \[0 \leqslant\dim_{\mathrm{F}}\mu = \dim_{\mathrm{F}}^0 \mu \leqslant\dim^\theta_{\mathrm{F}}\mu \leqslant\dim_{\mathrm{F}}^1 \mu = \dim_{\mathrm{S}}\mu.\] Moreover, for compactly supported measures it is continuous on the closed interval \([0,1]\). Importantly, it was proved in [28] that if \(\mu\) is compactly supported, then \[\label{bound} \dim^\theta_{\mathrm{F}}\mu \leqslant\dim_{\mathrm{F}}\mu + d \theta.\tag{7}\] Despite its recent inception, the Fourier spectrum has already seen several applications in both fractal geometry and harmonic analysis; see, for example, [26], [29] for applications to the distance set problem, [30] for applications to exceptional set estimates for orthogonal projections, and [31] where it was used to study the Fourier restriction problem (in the positive direction).

There are various enemies of Fourier decay. For example, if a measure is supported on a hyperplane then it cannot decay at all in directions in the orthogonal complement of the hyperplane. On the other hand, the arithmetic resonance in the middle third Cantor set prevents any measure supported on the Cantor set from decaying at all. Nevertheless, the Fourier spectrum captures more nuanced average decay in both of these settings. The goal now is to use the quantitative flatness framework described in the previous section to inhibit Fourier decay. This turns out to be more challenging than with the Fourier restriction problem, but we are able to make some progress, which will turn out to be useful in several specific settings later on. Our main interest is in bounding the Fourier dimension from above, but we will often state our results in terms of the Fourier spectrum since these statements contain the Fourier dimension estimates (at \(\theta=0\)) and contain strictly more information whenever we can beat the estimate \[\dim^\theta_{\mathrm{F}}\mu \leqslant\dim_{\mathrm{F}}\mu +d \theta\] which always holds.

2.4.2 Bounding the Fourier dimension via Fourier restriction↩︎

In this section we derive general upper bounds for the Fourier dimension and spectrum in terms of quantitative flatness, thus providing a Knapp style obstruction in the Fourier decay problem.

The Stein–Tomas restriction theorem ensures a range of \(q\) for which the \(E(2\to q)\) extension estimate holds in terms of the Fourier dimension and Frostman dimensions of a measure. The Frostman dimension of \(\mu\) with support given by \(X\) is defined by \[\begin{align} \dim_{\mathrm{Fr}}\mu = \sup \Big\{ s \geqslant 0 : \exists C>0 \;, \;\forall \;r>0 \; , \;\forall x \in \mathbb{R}^d\; , \; \mu(B(x,r)) \leqslant Cr^{s} \Big\} . \end{align}\] Therefore, since Theorem 3 uses \(\gamma\)-flatness to inhibit extension estimates, we may bound the Fourier dimension from above by appealing to \(\gamma\)-flatness.

One drawback of Corollary 8 is that it requires knowledge of the Frostman dimension of \(\mu\). We can use the general estimate \(2\alpha \geqslant\dim_{\mathrm{F}}\mu\) due to Mitsis [32] (see also Theorem 18) to get the following Frostman independent bound.

In order for the estimate from the above corollary to be non-trivial the \(\gamma\) needs to be rather small, certainly satisfying \(\gamma<\dim_{\mathrm{H}}\mu/(2d)\).

In [31] the Stein–Tomas estimate was generalised to use information from the Fourier spectrum rather than only the Fourier dimension. Following the same approach as Corollary 8 we get an estimate for the Fourier spectrum.

One interesting feature of Corollary 10 is that it is stronger than the estimate coming from Corollary 8 combined with basic information about the Fourier spectrum, see 7 . In particular, \[\dim^\theta_{\mathrm{F}}\mu \leqslant\dim_{\mathrm{F}}\mu + d \theta \leqslant\frac{2\gamma(d-\alpha) }{1-\gamma} + d \theta\] but the right hand side is a strictly worse estimate than Corollary 10, recalling that we are interested in the \(\alpha<d\) regime.

2.4.3 Bounding the Fourier dimension using flatness directly↩︎

If a measure is \(\gamma\)-flat, especially with anisotropic witnessing cuboids and \(\gamma<1\), then \(\mu\) must have large amounts of mass concentrated in relatively flat regions. This is an enemy of Fourier decay and so one might hope to use this information directly to bound the Fourier dimension from above, that is, without going via Fourier restriction. It turns out that for this we need the refined concept of \((\gamma,\upsilon)\)-flatness, but even in this setting we only have partial success. We are able to prove some estimates in certain restricted situations and we explain why something stronger cannot hold. The estimates we obtain here should be compared with those coming from restriction theory in Corollary 12. Sometimes one approach is better, sometimes the other is better, as we shall see later.

We begin with a heuristic discussion which is meant to hint at what one might hope to get from \((\gamma,\upsilon)\)-flatness. To this end, let \(\mu\) be a \((\gamma,\upsilon)\)-flat measure and \(Q_k\) be a sequence of witnessing cuboids. To simplify this heuristic discussion, assume all of the \(Q_k\) are centred at the origin. We will revisit the heuristic in full generality when we prove Theorem 14 later. Let \[\delta_k = \min_j |I_j^k|\] be the length of the shortest side of \(Q_k\) and note that \(\delta_k \to 0\) as \(k \to \infty\). We also have the estimates \[\mathcal{L}^d(Q_k) \gtrsim \delta_k^\upsilon\] and \[\mu(Q_k) \gtrsim \mathcal{L}^d(Q_k)^\gamma\] by assumption. Write \(Q_k^*\) for the dual cuboid as in the proof of Theorem 3. Consider the decomposition \[\mu = \mu|_{Q_k^c} + \mu|_{Q_k}\] and observe that for \(\xi \in Q_k^*\) \[\begin{align} \label{vgoodbound} \widehat{\mu|_{Q_k} }(\xi) = \int_{Q_k} e^{-2\pi i \xi\cdot x} \,d\mu(x) \in B\left( \mu(Q_k), \mu(Q_k)/10\right). \end{align}\tag{8}\] In particular, for \(\xi \in Q_k^*\) \[| \widehat{\mu|_{Q_k} }(\xi) | \approx \mu(Q_k) \gtrsim \delta_k^{\upsilon \gamma}.\] Since the diameter of \(Q_k^*\) is \(\approx \delta_k^{-1}\), this suggests a bound \(\dim_{\mathrm{F}}\mu \leqslant 2 \upsilon \gamma\) might be in play. Let \[A_k^* \subseteq Q_k^* \setminus B(0,\delta_k^{-1}/1000)\] be a cuboid with one side of length \(\approx \delta_k^{-1}\) and the other sides of lengths \(\approx 1\). We are searching for \(\xi\) with \(|\xi| \approx \delta_k^{-1}\) for which \[\left| \widehat{\mu } (\xi)\right| \gtrsim \mu(Q_k)\] and this is guaranteed if we can find \(\xi \in A_k^*\) satisfying \[\left| \widehat{\mu }(\xi) \right| \gtrsim \mu(Q_k).\] Indeed, if we have such \(\xi \in A_k^*\), then \[\left| \widehat{\mu } (\xi)\right| \gtrsim \mu(Q_k) \gtrsim \mathcal{L}^d(Q_k)^\gamma \gtrsim \delta_k^{\upsilon \gamma}\approx |\xi|^{-\frac{2\upsilon \gamma}{2}}\] and the bound \[\dim_{\mathrm{F}}\mu \leqslant 2\upsilon \gamma\] would follow. Therefore, suppose for all \(\xi \in A_k^*\), \[\left| \widehat{\mu}(\xi) \right| \leqslant\mu(Q_k)/10000.\] For this to be true, recalling 8 , we require that for all \(\xi \in A_k^*\), \[\widehat{\mu|_{Q_k^c} }(\xi) \in B\left(-\mu(Q_k), \mu(Q_k)/5\right)\] which seems like a very strong requirement. Indeed, it is asking for the near-constancy of \(\widehat{\mu|_{Q_k^c} }(\xi)\) for a very large set of \(\xi\). Any substantial oscillation in \(\widehat{\mu|_{Q_k^c} }\) (for example, a sign change) over this range of frequencies would kill this possibility immediately.

However, despite the above observations and heuristics, the desired bound \[\dim_{\mathrm{F}}\mu \leqslant 2 \upsilon \gamma\] is not true in general—not even close!

In Example 3 we could have taken \(\varepsilon=0\) but we included the \(\varepsilon>0\) to ensure that the diameters of the cuboids tended to zero. This is not necessary of course. In fact the hoped for bound can fail much more generally as the next example shows.

At this point the reader may be rightly suspicious of the previous two examples because they both had \(\gamma=1\). Recall that in Corollary 12 we necessarily needed \(\gamma<1\) to get something non-trivial and so perhaps adding the assumption \(\gamma<1\) can save the desired upper bound \(2 \upsilon \gamma\). Next we show that, unfortunately, this is not the case.

The above example is disappointing but, perhaps surprisingly, we can recover the bound \(\dim_{\mathrm{F}}\mu \leqslant 2 \upsilon \gamma\) in some fairly general situations. First, provided the product \(\upsilon \gamma\) is strictly less than 1, we can recover the desired bound.

The previous result follows from the more general result below.

It is interesting to compare Example 5 and Theorem 12. In Example 5, \(\upsilon \gamma > 1+\varepsilon t\) and this can be made arbitrarily close to 1. But for Theorem 12 to apply with cuboids with the anisotropy present in Example 5 we would need \(\upsilon \gamma <1\).

One of the main limitations of Theorem 12 is that we are forced to assume some isotropy in the cuboid \(Q_k\), that is, we are forced to have \(m\) ‘shortest sides’. This is needed to force the projection of \(Q_k\) into a ball. On the other hand, if we could control the measure of an anisotropic ball (e.g. an ellipsoid or cuboid) in terms of the Fourier dimension, then this problem would disappear. Unfortunately, we cannot do this in a sharp enough way to get anything useful. The best one can do directly is to cover the anisotropic ball with smaller isotropic balls but this is quite inefficient in general. We leave the reader to formulate estimates coming from this approach. On the other hand, if the Fourier dimension is so big that the projection is forced to have a continuous density, then the measure of anisotropic balls can be controlled in a precise and useful way. This leads to the following result.

It is clear from the formula in Corollary 8 that there is only non-trivial information when \(\gamma<1\) but it is less immediately obvious why \(\gamma<1\) is needed in Theorems 11 and 12. However, in (the more general) Theorem 12 the partial isotropy forces \(\upsilon \geqslant m\) and since we require \(\upsilon \gamma < m\) we must have \(\gamma<1\). In general we cannot expect any useful bounds to hold if \(\gamma \geqslant 1\). This can be seen by the Examples 3 and 4 above.

We conclude this section with another situation where we can obtain the desired upper bound. The drawback this time is that we require an awkward integrability assumption ?? which says that the measure decays away from the cubes witnessing the flatness. The aim was to give a non-Fourier analytic condition.

2.5 \(L^2\)-flattening↩︎

\(L^2\)-flattening is an important property which a measure can satisfy which has many useful applications, see [34]–[37]. Roughly speaking, the idea is that convolution with an \(L^2\)-flattening measure quantitatively improves the \(L^2\) dimension, see [38]. Importantly, \(L^2\)-flattening does not require (but is implied by) positive Fourier dimension (\(L^\infty\) decay).

We say a compactly supported Borel measure \(\mu\) on \(\mathbb{R}^d\) is \(L^2\)-flattening if for all \(\varepsilon>0\), there exists \(p>1\) such that for all \(R>0\) \[\| \widehat{\mu} \|_{L^p(B(0,R))}^p = O_\varepsilon(R^\varepsilon).\] The following observation shows that \(L^2\)-flattening is completely characterised by the Fourier spectrum. Later we will use this characterisation to connect \(L^2\)-flattening to quantitative flatness. Here we write \(A(\theta) \sim B(\theta)\) as \(\theta \to 0\) if \(A(\theta)/B(\theta) \to 1\) as \(\theta \to 0\).

This characterisation perhaps suggests a more thorough analysis of similar flattening properties. For example, do the higher order terms in \(\dim^\theta_{\mathrm{F}}\mu\) as \(\theta \to 0\) play a role? Moreover, when \(\mu\) is not \(L^2\)-flattening, concavity of the Fourier spectrum implies that \[\dim^\theta_{\mathrm{F}}\mu \geqslant\theta \dim_{\mathrm{S}}\mu\] holds for all \(\theta\). Therefore, one might ask whether an intermediate asymptotic, such as \[\dim^\theta_{\mathrm{F}}\mu \sim u \theta\] as \(\theta \to 0\) for some \(u \in (\dim_{\mathrm{S}}\mu , d)\), is useful or whether we always need \(u=d\). Further, we wonder if there are interesting classes of measures which satisfy such an intermediate asymptotic.

Since \(L^2\)-flattening occurs whenever \(\dim_{\mathrm{F}}\mu >0\), we can only inhibit \(L^2\)-flattening in the very special case when we can show \(\dim_{\mathrm{F}}\mu =0\). But even in this case we have to do more and show that the Fourier spectrum is not asymptotically as large as possible (that is, does not satisfy \(\dim^\theta_{\mathrm{F}}\mu \sim d\theta\)). Perhaps surprisingly, we achieve the latter whenever we achieve the former. We also achieve this in a uniform and quantitative sense by proving that the Fourier spectrum is bounded above by \((d-1/2)\, \theta\).

The upper bound of \((d-1/2)\) for the Sobolev dimension in Theorem 16 is intriguing and a first question is whether it is optimal. In fact, it is not and we prove something more general below. Note that bounding the Sobolev dimension above by \((d-1)\) does not improve on the upper bound for the Fourier spectrum in Theorem 16 apart from for \(\theta\) close to 1 and this is not what is relevant in the characterisation of \(L^2\)-flattening. We do not know if \((d-1/2) \, \theta\) is the optimal upper bound for the Fourier spectrum near zero, noting that \(0\)-flat examples with Fourier spectrum equal to \((d-1)\, \theta\) are easily constructed, see below.

3 Background on fractal geometry and dimension theory↩︎

In this section we recall several further notions of fractal dimension, of both sets and measures, which we will use throughout the rest of the paper. The goal now is to use familiar fractal geometric concepts to conclude \(\gamma\)-flatness and \((\gamma,\upsilon)\)-flatness and then to apply the general results from the previous section in various settings of interest.

First, the upper box dimension of a bounded set \(X \subseteq \mathbb{R}^d\) is defined by \[\overline{\dim}_{\mathrm{B}}X = \limsup_{r \to 0} \frac{\log N_r(X)}{-\log r}\] where \(N_r(X)\) is defined to be the smallest number of sets of diameter \(r\) required to cover \(X\). Similarly, the lower box dimension \(\underline{\dim}_{\mathrm{B}}X\) is defined by replacing the limsup with a liminf. If the upper and lower box dimensions coincide, then we refer to the common value as the box dimension and denote it by \(\dim_{\mathrm{B}}X\).

The Assouad spectrum, introduced in [40], interpolates between the upper box dimension and the Assouad dimension in a meaningful way. It provides a parametrised family of dimensions by fixing the relationship between the two scales \(r<R\) used to define Assouad dimension. Studying the dependence on the parameter within this family thus yields finer and more nuanced information about the local structure of the set. For example, one may understand which scales ‘witness’ the behaviour described by the Assouad dimension. For \(\theta \in (0,1)\), the Assouad spectrum of \(X\) is given by \[\begin{align} \dim^\theta_{\mathrm{A}}X = \inf \Bigg\{ s \geqslant 0 : \exists C>0 \;, \;\forall \;0<r<1 \; , \;\forall x \in F \; , \; N_r(B(x,r^{\theta}) \cap X) \leqslant C \left(\frac{r^{\theta}}{r} \right)^{s} \Bigg\} . \end{align}\] See [16] for more background and basic properties of the Assouad and lower spectra. It was shown in [40] that, for a bounded set \(F \subseteq \mathbb{R}^d\), \[\begin{align} \overline{\dim}_{\mathrm{B}}X &\leqslant\dim^\theta_{\mathrm{A}}X \leqslant\frac{\overline{\dim}_{\mathrm{B}}X}{1-\theta} . \label{basicbound} \end{align}\tag{11}\] In particular, \(\dim^\theta_{\mathrm{A}}X \to \overline{\dim}_{\mathrm{B}}X\) as \(\theta \to 0\). The limit \(\lim_{\theta \to 1} \dim^\theta_{\mathrm{A}}X\) exists and equals the quasi-Assouad dimension, which is often but not always equal to the Assoaud dimension. The quasi-Assouad dimension was introduced in [41].

The upper box dimension of \(\mu\) with bounded support is defined by \[\begin{align} \overline{\dim}_{\mathrm{B}}\mu = \inf \Big\{ s \geqslant 0 : \exists C>0 \;, \;\forall \;0<r< 1 \; , \;\forall x \in X \; , \; \mu(B(x,r)) \geqslant Cr^{s} \Big\} \end{align}\] and the lower box dimension of \(\mu\) is given by \[\begin{align} \underline{\dim}_{\mathrm{B}}\mu = \inf \Big\{ s \geqslant 0 : \exists C>0 \; , \;\forall r_0>0 \;, \; \exists \;0<r<r_0 \;, \;\forall \;x\in X \; , \; \mu(B(x,r)) \geqslant Cr^{s} \Big\}. \end{align}\] If \(\overline{\dim}_{\mathrm{B}}\mu = \underline{\dim}_{\mathrm{B}}\mu\), then we refer to the common value as the box dimension of \(\mu\), denoted by \(\dim_{\mathrm{B}}\mu\). These definitions of the box dimension of a measure were introduced in [42]. It was shown that the bounds \[\overline{\dim}_{\mathrm{B}}X \leqslant\overline{\dim}_{\mathrm{B}}\mu\] and \[\underline{\dim}_{\mathrm{B}}X \leqslant\underline{\dim}_{\mathrm{B}}\mu\] always hold and these inequalities are sharp in the sense that \(X\) always supports a measure which attains equality in either of the above. In this way, the box dimensions of a set are characterised by the box dimensions of measures they support.

The upper box dimension of a measure is dual to the Frostman dimension, \(\dim_{\mathrm{Fr}}\mu\) which we have already met. In general, \[\dim_{\mathrm{Fr}}\mu \leqslant\dim_{\mathrm{S}}\mu \leqslant\dim_{\mathrm{H}}\mu \leqslant\dim_{\mathrm{H}}X \leqslant\underline{\dim}_{\mathrm{B}}X \leqslant\overline{\dim}_{\mathrm{B}}X \leqslant\overline{\dim}_{\mathrm{B}}\mu.\]

4 Quantitative flatness from geometry↩︎

4.1 Basic estimates↩︎

In this section we present some very simple applications of our ‘quantitative flatness’ approach. In this way we will see some well-known and straightforward facts recovered in this context. This serves as a warm up for later applications and as a sanity check against known constraints.

One can see from Theorem 3 that in order to obtain an obstruction to Fourier restriction, the aim is to find places where a lot of \(\mu\) mass is found in a relatively small volume cuboid. Of course, one can always take the cuboid to be a cube and the mass is then controlled by the Frostman dimension. This basic application generalises a known obstruction in the \(p=2\) case. More specifically, we recover a basic threshold for restriction due to Hambrook–Łaba [43] and also an estimate for the Fourier dimension due to Mitsis [32].

A trivial but important observation which is in some sense an extreme special case of quantitative flatness is that if a measure in \(\mathbb{R}^d\) is supported in an affine hyperplane (or merely gives positive mass to a hyperplane), then there are no non-trivial restriction estimates and the Fourier dimension is necessarily zero. These observations follow easily from Theorem 3 and Corollary 8 but we also obtain the following more general observation which says that being sufficiently concentrated near an affine hyperplane gives the same conclusion.

4.2 \(C^2\) surfaces: using curvature↩︎

Many of the key examples in Fourier restriction theory are smooth manifolds, such as the sphere, cone, or hyperboloid. It is also of interest to study smooth manifolds in general, see, for example, [10], [44]–[46]. Here we show how quantitative flatness can be used in this setting by leveraging (lack of) curvature in the manifold.

We consider general \(C^2\) manifolds \(X \subseteq \mathbb{R}^d\) of dimension \(s \in \{1, \dots, d-1\}\). Such a manifold comes with a natural surface measure, which is the analogue of \(s\)-dimensional Lebesgue measure on the surface and these are our main measures of interest. A point \(x \in X\) is said to be of rank \(k\) if there are \(k\) non-zero principal curvatures and \(s-k\) principal curvatures equal to zero. Here \(s-k\) is the nullity. Moreover, we say a point \(x \in X\) is of ambient rank \(k\) if \[r^*(x) := \limsup_{y \to x} r(y) = k\] where \(r(y)\) denotes the rank of \(y\). We say \(n^*(x) = s-r^*(x)\) is the ambient nullity of \(x\). Then \(r^*(x) \geqslant r(x)\) for all \(x\) but this inequality can be strict. For example, \[X = \{(x,x^2): x \in [-1,1]\}\] is a \(C^2\) manifold of dimension 1 in the plane. The rank of every point is maximal (equal to 1) apart from at \((0,0)\) where it is 0. However, the ambient rank of every point is 1. Of course, if \(X\) is of constant rank, that is, \(r(x)\) is the same for all \(x \in X\), then ambient rank and rank coincide at all points.

There is considerable interest in both the Fourier restriction and Fourier decay problem for these surface measures. The Fourier restriction theory is fairly classical but the Fourier decay problem has seen a lot of attention recently. For example, it was proved in [2] that any smooth constant rank \(k\) hypersurface (that is, of co-dimension \(d-s=1\)) has Fourier dimension \[\dim_{\mathrm{F}}X = k.\] It is an interesting problem to relax the assumptions, especially the constant rank and co-dimensionality assumptions. We make progress in both of these directions.

The previous result established that the Fourier dimension of the surface measure on a \(C^2\) manifold of arbitrary dimension and variable rank is bounded above by the smallest ambient rank of a point. This estimate is as good as one can hope for in this level of generality. In particular, the ambient rank cannot be replaced by the rank, which is what one might have naively conjectured. Indeed, let us reconsider the simple example \[X = \{(x,x^2): x \in [-1,1]\}.\] Here the minimal rank is zero but the minimal ambient rank is 1. However, it follows easily from a stationary phase argument that the Fourier dimension of the surface measure is, in fact, 1.

Next we consider arbitrary measures on \(X\) and the Fourier dimension of the manifold itself. In the constant rank \(r\) case, it is natural to conjecture that \[\label{surfaceconjecture} \dim_{\mathrm{F}}X \leqslant r.\tag{12}\] In general nothing more can be said, because if \(s \leqslant d-2\), then it is possible for \(X\) to be contained in a hyperplane, even when the rank is maximal, in which case the Fourier dimension is zero. However, if \(s=d-1\), that is, we are in the setting of co-dimension 1 hypersurfaces, then it follows by a straightforward stationary phase argument that the Fourier dimension of a small piece of the surface measure is at least \(r\), which gives the corresponding lower bound. The main interest is therefore in the upper bound 12 , but bounding the Fourier dimension of \(X\) above requires bounding the Fourier dimension uniformly above for all measures on \(X\) and this is difficult. The above conjecture 12 has recently been verified in the co-dimension 1 case, that is, \(s=d-1\) [2]. In the following subsection we provide an alternative approach to this problem and show that if the dimension of the manifold is small enough, then 12 is not optimal and can be improved in certain cases. In the following, we verify 12 in the constant curvature arbitrary co-dimension case under the assumption \(r/2+s<d\). Since there is nothing to prove when \(r=s\) because then \(\dim_{\mathrm{F}}X \leqslant\dim_{\mathrm{H}}X = s\), we may assume \(r \leqslant s-1\). Therefore, we resolve the (constant rank) conjecture in all cases when \[s<\frac{2d+1}{3}.\] In the variable rank case, we show the largest ambient rank is an upper bound. At first sight this might seem sub-optimal because in Theorem 20 we proved that the smallest ambient rank is an upper bound for the Fourier dimension of the surface measure. However, when one is considering the manifold itself, in the variable rank case one may choose measures which avoid the locations with lowest rank and so instead the largest rank becomes the relevant quantity. For example, suppose \(X\) contains a genuinely flat plane of co-dimension 1 and a curved part (where we glue the pieces together in a \(C^2\) way). Then the smallest ambient rank is zero, but the Fourier dimension can be strictly positive by witnessing this via measures supported only on the curved part.

4.3 Smooth surfaces: using dimensionality↩︎

By following an approach of Gressman [17] we next provide an alternative strategy for bounding the Fourier dimension of smooth submanifolds where we leverage low dimensionality, rather than low rank as in the previous subsection. We arrive at some surprising conclusions, including that smooth curves in \(\mathbb{R}^d\) have Fourier dimension at most \(4/(d+1)\).

We again consider \(s\)-dimensional submanifolds in \(\mathbb{R}^d\). Gressman [17] introduced the quantity \(H=H(s,d)\) which he called the homogeneous dimension and is equal to the smallest positive integer which can be written as the sum of the degrees of some collection of \(d\) distinct, nonconstant monomials in \(s\) variables. Interpreting the main result of [17] in our language, we get the following.

Combining the previous result with Corollary 8, Corollary 10 and Mitsis’ Frostman dimension bound (see Corollary 9) we get an alternative way to bound the Fourier dimension of a submanifold which is especially good when \(s\) is small. Note that there are no assumptions regarding rank or nullity in what follows.

The bound for the Fourier dimension of the manifold \(X\) itself is not useful when \(s\) is large. Indeed, \(H(d-1,d) = d+1\) since there are \(d-1\) distinct linear monomials and the next smallest monomial is necessarily of degree 2, and so the bound exceeds \(s = \dim_{\mathrm{H}}X\) when \(s=d-1\). However, when \(s\) is small, something interesting happens. We first consider curves where computing \(H(s,d)\) is straightforward.

Next we consider higher dimensional submanifolds where estimating \(H(s,d)\) becomes a little more difficult. Again we see that when \(d\) is sufficiently large an \(s\)-dimensional manifold cannot be Salem in \(\mathbb{R}^d\) and our bound for the Fourier dimension goes to zero as \(d \to \infty\).

4.4 Fractal surfaces↩︎

In this section we remain in the setting of surfaces, but we drop all smoothness assumptions. We consider general \((d-k)\)-dimensional topological manifolds which we assume are the graphs of Hölder functions \(f: [0,1]^{d-k} \to \mathbb{R}^k\) where the graph is defined by \[G_f= \{(x,f(x)) : x \in [0,1]^{d-k} \} \subseteq \mathbb{R}^d.\] Then we replace the surface measure by the lift of Lebesgue measure on \([0,1]^{d-k}\) to the graph. Without derivatives, the standard Knapp style examples no longer apply and instead we prove quantitative flatness using other geometric properties of the surface including Hölder continuity and fractal dimensions. First, we obtain a basic estimate coming directly from the Hölder exponent. We get useful information in the context of restriction and \(L^p\)-improving properties, but not in terms of Fourier dimension. The special case \(k=1\) of this estimate was already observed in [48].

Our next goal is to beat the previous estimate by replacing the ‘vertical cuboids’ from the previous result with ‘horizontal cuboids’ which take advantage of fractal features of the surface. We can obtain such fractal information by considering the Assouad spectrum of ‘level sets’ \(f^{-1}(z)\), for \(z \in \mathbb{R}^k\) in the range. For complicated functions \(f^{-1}(z) \subseteq [0,1]^{d-k}\) is often a fractal set and may have non-trivial dimensional information. In particular, large level sets coupled with the Hölder condition guarantee lots of mass is trapped in a relatively flat cuboid. We assume \(\theta \in [0,\alpha]\) to force the cuboids we use in the proof to be horizontally flat and so genuinely distinct from the vertical cuboids considered in Theorem 26. Otherwise Theorem 26 will be better, because the Hölder condition traps the whole surface in the cuboid without appealing to dimension theory of level sets.

Theorem 27 already beats Theorem 26 in the \(\theta=0\) case whenever there is a level set with upper box dimension \[> \frac{(d-k)^2}{d-k+k\alpha}\] and it can beat it with even less restrictive bounds on the Assouad spectrum. The Fourier dimension bound is also interesting. In this setting \[\dim_{\mathrm{F}}\mu \leqslant\dim_{\mathrm{F}}\mathcal{L}^{d-k}\vert_{[0,1]^{d-k}} = 2\] always holds since projection cannot decrease Fourier dimension. Therefore we are interested in conditions which beat this estimate. At \(\theta=0\), this will be achieved whenever \[\overline{\dim}_{\mathrm{B}}f^{-1}(z) >d-k-\frac{k\alpha}{k+1}\] for some \(z\), which can be satisfied since the level sets live in \((d-k)\)-dimensional space. More generally, one may derive a condition in terms of the Assouad spectrum and which \(\theta\) is optimal is not pre-determined. That said, the bound at \(\theta=\alpha\) is never better than 2 and so one looks for intermediate \(\theta\)s.

Concerning the second (conditional) Fourier dimension bound, note that it is always better than the first when it applies. Moreover, when \(k \geqslant d/(1+\alpha)\) it always applies.

Next we apply the converse of Theorem 27 to get uniform information about the box dimension of the level sets of the graphs of stochastic processes. These estimates are non-optimal, but at least give non-trivial estimates via a rather different approach.

It is well-known that almost surely for Lebesgue positively many \(z\) the level sets of classical Brownian motion have Hausdorff dimension \(1-H\). The fact that we obtain \(1-H/3\) as an upper bound for all level sets and with Hausdorff dimension replaced by the a priori larger upper box dimension is perhaps surprising to the non-expert. That said, it is known that \(1-H\) is, in fact, also an upper bound for the upper box dimension of all level sets, but this requires non-trivial ideas from stochastic analysis to prove, see [52], [53].

4.5 Tube null sets↩︎

A set \(X\) in \(\mathbb{R}^2\) is called tube null if it can be covered efficiently by tubes. The property of being tube null is quite subtle and relates to several different problems, see [10], [54]–[56]. Here a tube is the \(\delta\)-neighbourhood of a line and the width of the tube is \(\delta\). More precisely, \(X\) is tube null if for all \(\varepsilon>0\) there exists a countable collection of tubes \(T_i\) of width \(|T_i|\) such that \[\sum_i |T_i| < \varepsilon.\] A tube null set must have Lebesgue measure zero but it can have Hausdorff dimension 2. Any set with Hausdorff dimension \(<1\) is tube null. More generally, for \(1 \leqslant k<d\) integers and \(s \in (0,1]\), we say a set \(X \subseteq \mathbb{R}^d\) is \((s,k)\)-tube null if for all \(\varepsilon>0\) there exists a countable collection of affine subspaces \(\{L^i\}_i\) of dimension \(k\) and a collection of corresponding widths \(\delta_i\) such that \[X \subseteq \bigcup_i (L^i)_{\delta_i}\] and such that \[\sum_i \delta_i^{s(d-k)} < \varepsilon.\] The traditional definition of a set in \(\mathbb{R}^d\) being tube null now coincides with it being \((1,1)\)-tube null and being \((s,1)\)-tube null with \(s<1\) is a strictly stronger condition meaning that the collection of tubes is ‘even smaller’. We are unaware of this refined notion (with \(s<1\)) appearing in the literature but it is so natural that we suspect it has been considered before. We first prove some basic estimates concerning \((s,k)\)-tube null sets.

Tube nullity forces large amounts of the set to be contained in tubes, which are (infinite extensions of) relatively flat cuboids. We can therefore transfer information about tube nullity into our quantitative flatness framework and derive information useful for Fourier restriction and dimension estimates.

Concerning the Fourier dimension bounds we were in the fortunate position above to be able to apply Theorem 13. However, we could also apply Corollary 8. We get that if \(\alpha\) is the Frostman dimension of \(\mu\) then \[\dim_{\mathrm{F}}\mu \leqslant\frac{2s(d-\alpha) }{1-s}.\] However, this bound is always worse than the bound \(2s(d-k)\) obtained above. This is because by Theorem 29 \(\alpha \leqslant\dim_{\mathrm{H}}X \leqslant k+s(d-k)\).

Next we revisit a some simple examples in the tube nullity context.

The above approach does not work well for spheres in higher dimensions, since they are not tube null in a useful sense. Indeed, consider \[\approx \delta^{-\frac{d-1}{2}}\] many uniformly spaced tangent planes to \(S^{d-1} \subseteq \mathbb{R}^d\). The associated \(\delta\)-tubes cover \(S^{d-1}\) and \[\delta^{-\frac{d-1}{2}} \delta^s \to 0\] provided \(s>(d-1)/2\) but this is trivial for \(d \geqslant 3\).

Again, the above strategy does not work in higher dimensions (this time \(d \geqslant 4\)) because then the cones fail to be tube null in a useful sense. It does work for the ‘cone’ in the plane, but this is a trivial example because it is just two line segments.

4.6 Flatness via large slices↩︎

An important problem in geometric measure theory is to understand the ‘slices’ of a fractal and, in particular, how one can relate the dimensions of the slices to the dimensions of the original set. Here a slice is the intersection of the set with an affine subspace. We write \(A(d,k)\) for the affine Grassmannian consisting of all \(k\)-dimensional affine subspaces \(V= V'+z\) for some \(V' \in G(d,k)\) and \(z \in V^{\perp}\).

If a set has a large slice, then one might expect lots of the set to be concentrated near the associated affine subspace and that this provides quantitative flatness. However, this is not the whole story as having a large slice is not enough to control the measure near the slice. Instead, the strategy is to couple the information coming from the slice (which we quantify using the Assouad spectrum) with global control of the measure on small balls coming from the box dimension (of the measure) to get what we need.

Given the previous result, it is natural to ask whether non-trivial estimates come from Marstrand’s slicing theorem which ensures that all sufficiently large sets must have some (in fact many) large slices. However, the answer is no and, in fact, this is to be expected since the bounds would apply too generally. In other words, to get non-trivial information we need atypically large slices.

In order to obtain strong results we are looking for a measure with small box dimension supported on a set with a slice with large Assouad spectrum. In the extreme case when the former is as small as possible and the latter is as large as possible we obtain optimal estimates. Recall that we may always choose \(\mu\) on \(X\) with \(\overline{\dim}_{\mathrm{B}}\mu = \overline{\dim}_{\mathrm{B}}X\) and so if a set has a slice with maximal Assouad spectrum then there always exists a measure—a very natural measure—on the set with Fourier dimension 0 and no restriction estimates. The following result is a direct corollary of Theorem 31 and Theorem 16.

For certain examples we may find that atypically large slice exists by construction. We state a simple corollary which applies to two well-known dynamically defined fractals.

4.7 Nonlinear slices↩︎

The results in the previous section say that if an affine subspace intersects the support of a measure in a set with large Assouad spectrum then the measure is forced to be quantitatively flat. A natural extension of this is to replace the affine subspace with a (nonlinear) \(k\)-dimensional surface with suitable smoothness. This is not just a routine extension because the same results clearly cannot hold. If the surface is curved enough then this can preserve Fourier decay even for measures which are very close to (or even completely contained in) the surface. That is, we will necessarily see different features appearing in the estimates coming from the (potential) curvature. We write \(a \wedge b = \min\{a,b\}\).

It is interesting to compare Theorem 34 with Theorem 31. One notable difference is that Theorem 34 does not give anything useful in the case \(\theta=0\). This is because when \(\theta\) is very small, the (potential) curvature of \(S\) forces \(Q\) to become quantitatively less flat. Moreover, we do not get a conclusion of the form: a measure with box dimension \(k\) fully supported on a set which contains a piece of a \(k\)-dimensional surface has Fourier dimension zero. But this is simply not true, for example the surface measure on a sphere. Nevertheless, we can still obtain an analogue of Corollary 33 but w are forced to use larger \(\theta\). In fact, the bound obtained is optimised at \(\theta=1/2\) (provided \(\overline{\dim}_{\mathrm{B}}\mu \leqslant d\)) but we see a loss compared with the estimate in Corollary 33 and this can be interpreted as coming from potential curvature.

The bound \(2s-k\) is sharp since for the Surface measure on the sphere \(S^{d-1} \subseteq \mathbb{R}^d\) and \(S\) the sphere itself, \(s=k=d-1\) and \(\dim_{\mathrm{F}}\mu = d-1\).

4.8 Flatness via large projections↩︎

Another central problem in geometric measure theory is to understand how the dimensions of fractal sets behave under orthogonal projection. The seminal Marstrand–Mattila projection theorem tells us that the projection of a Borel set \(X \subseteq \mathbb{R}^d\) onto a \(k\)-dimensional subspace has Hausdorff dimension \(\dim_{\mathrm{H}}X \wedge k\) almost surely with respect to the Grassmannian measure. For \(V \in G(d,k)\), we write \(P_V\) for orthogonal projection of \(\mathbb{R}^d\) onto \(V\).

Having a projection with large Hausdorff dimension does not give us quantitative flatness but, on the other hand, a projection with large Assouad spectrum means that a large amount of the set has projected into a small ball and the preimage of this small ball under the projection is a thin tube. This information, one again coupled with information coming from the box dimension of the relevant measure which ensures that all balls carry a certain amount of mass, forces a lot of mass into a small tube and this is the quantitative flatness we need. This idea is made precise in the following result.

In order to obtain strong results we are looking for a measure with small box dimension supported on a set which has a projection with large Assouad spectrum. In the extreme case, when the former is as small as possible and the latter is as large as possible, we obtain optimal estimates. Recall that we may always choose \(\mu\) fully supported on \(X\) with \(\overline{\dim}_{\mathrm{B}}\mu = \overline{\dim}_{\mathrm{B}}X\) and so if a set has a projection with maximal Assouad spectrum then there always exists a very natural measure on the set with Fourier dimension 0 and no restriction estimates.

Next we present a modest but perhaps surprising application to a well-known problem concerning the dimensions of self-similar sets. Recall that a self-similar set is a non-empty set \(X \subseteq \mathbb{R}^d\) which is invariant under an iterated function system (IFS) of similarity maps. See [14], [15] for more details on self-similar sets and their dimension theory. It is known that if the open set condition is satisfied, then the Assouad, box and Hausdorff dimensions of \(X\) coincide and are given by a simple closed formula (the Hutchinson–Moran formula). However, if the open set condition fails then there are two rather different regimes. If the weak separation condition holds, then these three dimensions coincide, but if the weak separation condition fails, then the Assouad dimension may be strictly larger. Nevertheless, it is widely believed that without any separation assumptions the Assouad spectrum remains equal to the box dimension for all \(\theta\). This is known for certain cases, for example, if the self-similar set is in the line, the defining IFS is algebraic, and there are no exact overlaps. A modest step in the direction of this conjecture would be to bound the Assouad spectrum above by something strictly smaller than the general upper bound coming from the box dimension, that is, \[\dim^\theta_{\mathrm{A}}X \leqslant\frac{\overline{\dim}_{\mathrm{B}}X}{1-\theta}.\] We achieve this using our quantitative flatness assumption combined with known results about the \(L^2\)-flattening nature of self-similar measures. The result applies to self-similar sets whose Hausdorff dimension is given by the Hutchinson–Moran formula. The interesting case is when the open set condition fails and so we are interested in the situation when there are no exact overlaps and the exact overlaps conjecture holds. We can handle more general classes than this with the following method but we leave the details to the reader. The important thing is that the dimension is obtained as the sequence of solutions to Hutchinson–Moran formulae applied to higher iterates of the defining IFS which account for exact overlaps.

The result below can be obtained by more direct methods, see [16]. We include the result because its proof involves a simple and novel synthesis of modern tools.

5 Examples and specific constructions↩︎

5.1 Self-affine sets↩︎

One of the most important ways to generate fractals sets, especially with a dynamical flavour, is via iterated function systems (IFSs). An IFS is a (usually finite) collection of contractions acting on a common compact set \(K \subseteq \mathbb{R}^d\). Such a collection defines a unique attractor which is often fractal. Here we focus on the case where the contractions are affine maps \(\{x \mapsto A_i(x) + t_i\}_{i\in \mathcal{I}}\) with each \(A_i\) an invertible matrix and each \(t_i\) a translation. We write \(S_i\) for the map \(S_i(x) = A_i(x)+t_i\). The associated attractor or self-affine set is the unique non-empty compact set \(X \subseteq \mathbb{R}^d\) satisfying \[X = \bigcup_{i \in \mathcal{I}} \left(A_i(X)+t_i\right).\] There is a natural surjection from the shift space \(\mathcal{I}^\mathbb{N}\) to the attractor (the coding map) and the dynamics generated by the IFS is closely related to the dynamics on the shift space given by the left shift. We consider (in the first instance not necessarily invariant) measures \(\nu\) on the shift space and their pushforwards onto \(X\) which we denote by \(\mu\). These are our measures of interest. For example, if \(\nu\) is a Bernoulli measure, then the associated \(\mu\) is a self-affine measure. We refer the reader to [14], [15] for more background on self-affine sets and measures.

We use standard symbolic notation to describe the \(\mu\). We write \(\mathcal{I}^*\) for the set of all finite words over \(\mathcal{I}\) and for \(\mathbf{i}=(i_1, i_2, \dots, i_k) \in \mathcal{I}^*\) we write \(1 \geqslant\alpha_1(\mathbf{i}) \geqslant\alpha_2(\mathbf{i}) \geqslant\cdots \geqslant\alpha_d(\mathbf{i}) >0\) for the singular values of the matrix \(A_\mathbf{i} = A_{i_1} \circ \cdots \circ A_{i_k}\). We also write \([\mathbf{i}] \subseteq \mathcal{I}^\mathbb{N}\) for the cylinder consisting of all infinite words beginning with \(\mathbf{i}\). Note that the image of \([\mathbf{i}]\) under the coding map in \(X\) is contained in the geometric cylinder \(S_\mathbf{i}(X)\) where \(S_\mathbf{i} = S_{i_1} \circ \cdots \circ S_{i_k}\). Once again, we write \(a \wedge b = \min\{a,b\}\).

In most situations, the geometric cylinders \(S_\mathbf{i}(X)\) become very stretched and flattened as the length of \(\mathbf{i}\) increases. We use this to force quantitative flatness of the relevant measure.

The previous theorem was rather general since it concerned any measure \(\nu\) and we were forced to carefully optimise the product \(\upsilon \gamma\) instead of optimising \(\gamma\) and \(\upsilon\) separately. Next we provide some more digestible statements. If the measure \(\nu\) is ergodic, then we know the \(\nu\)-typical behaviour of the relevant limits in the above result. This allows us to give cleaner bounds in terms of well-known quantities, although these bounds may not be the best possible.

The approach to bound the Fourier dimension of an ergodic measure on a self-affine set based on Theorem 13 works best in the plane (\(d=2\)) because in general all of the singular values or Lyapunov exponents of \(\mu\) are different (that is, we cannot expect to do better than \(m=1\)). For this reason we now specialise to the planar case and we consider the situation where we know the Hausdorff dimension of the measure. Higher dimensional analogues are also possible but we leave these to the reader. Jordan–Pollicott–Simon [57] introduced the Lyapunov dimension of \(\mu\) in \(\mathbb{R}^2\) as \[\frac{H(\mu)}{\chi_1(\mu)} \wedge \frac{H(\mu)+\chi_2(\mu)-\chi_1(\mu)}{\chi_2(\mu)}\] and proved that this is always an upper bound for the Hausdorff dimension of \(\mu\). However, in many cases the Hausdorff and Lyapunov dimensions coincide, see [58].

The following very special case of Theorem 39 and Corollary 40 gives simple conditions in terms of familiar quantities which ensure that a self-affine measure is not Salem.

5.2 Parabolic Julia sets↩︎

Let \(T\) be a rational map of degree at least 2 acting on the Riemann sphere \(\mathbb{C} \cup \{\infty\}\) and let \(J_T\) be the closure of the repelling periodic points of \(T\). Provided \(J_T\) is not the whole sphere, we may assume it is a compact subset of \(\mathbb{C}\) (via conjugation by an appropriate Möbius transformation which sends a point not in \(J_T\) to the point at infinity) and so we may identify \(J_T\) with a compact set in \(\mathbb{R}^2\). The set \(J_T\) is called the Julia set of \(T\) and is often a beautiful fractal set. Julia sets are one of the most well-studied and important classes of dynamically defined fractal. We refer the reader to [59], [60] for more background on the dynamics of rational maps and Julia sets.

We specialise further to the case of parabolic Julia sets which are those containing no critical points of \(T\) but at least one parabolic point, where a parabolic point is a fixed point \(\omega \in \mathbb{C}\) for \(T\) such that \(|T'(\omega)|=1\). Then \(T\) admits an expansion \[T(z) = z+a(z-\omega)^{p(\omega)+1}+ \dots\] in a neighbourhood of \(\omega\) where \(p(\omega) \in \mathbb{N}\) is the petal number of \(\omega\). The petal numbers play an important role in the dynamics of \(T\), especially near the associated parabolic points \(\omega\). The Hausdorff and box dimensions of a parabolic Julia set are given by an explicit pressure formula and we denote this value by \(h\). Note that \[\frac{p(\omega)}{1+p(\omega)}<h<2\] for all parabolic points \(\omega\), see [61]. For a parabolic Julia set, there exist natural \(h\)-conformal measures \(\mu\) on \(J_T\) and these are our measure of interest, see [60], [62]. Such a \(\mu\) is a measure of maximal dimension, that is, \(\dim_{\mathrm{H}}\mu = h = \dim_{\mathrm{H}}J_T\).

Leclerc [9] recently proved that hyperbolic Julia sets have positive Fourier dimension, but the precise value is non-quantitative and hard to pin down. We are able to use the dynamics near parabolic points to force mass into long thin rectangles, thereby obtaining quantitative flatness estimates.

The upper bound for the Fourier dimension in Theorem 42 (3) approaches zero as \(h\) approaches \[\max_\omega \frac{p(\omega}{1+p(\omega)}.\] This is a (strict) general lower bound for \(h\) but it is sharp and so by taking \(h\) arbitrarily close to this limit we can exhibit examples where the Fourier dimension is arbitrarily small while the Hausdorff dimension remains bounded away from zero.

The estimate in the case \(p=2\) in Theorem 42 (1) is always a non-trivial estimate since \(h<2\) and so the right hand side is \(>2\) but, moreover, it is always better than the bound \[q \geqslant\frac{4}{\dim_{\mathrm{Fr}}\mu}\] coming from Theorem 18. This can be checked by using the additional information that \[\label{frostjulia} \dim_{\mathrm{Fr}}\mu = \min\left\{h,h+(h-1)\max_\omega p(\omega)\right\}\tag{13}\] which follows from the global measure formula.

We obtained the upper bound for the Fourier dimension in Theorem 42 from Theorem 12 but we can also obtain an estimate from Corollary 8 using the Frostman dimension formula 13 above. Interestingly, the alternative bound coming from Corollary 8 is the same as the one we obtained in Theorem 42 in the case \(h\geqslant 1\) but for \(h<1\) it is strictly worse.

5.3 Kleinian limit sets↩︎

Let \(\Gamma\) be a non-elementary geometrically finite Kleinian group acting on \((d+1)\)-dimensional hyperbolic space with \(d \geqslant 1\) an integer. Assume \(\Gamma\) contains at least one parabolic element. Let \(L(\Gamma)\) denote the limit set which is the closure of the hyperbolic fixed points of \(\Gamma\). We assume that \(L(\Gamma)\) is not the whole boundary and therefore we may assume that \(L(\Gamma)\) is a compact subset of \(\mathbb{R}^d\), that is, we use the upper half space model of hyperbolic space. The limit set is often a beautiful fractal set which encodes many dynamical features of the action of \(\Gamma\). We refer the reader to [64]–[66] for more background on Kleinian group actions and limit sets.

An element of \(\Gamma\) is called parabolic if it fixes precisely one point on the boundary of hyperbolic space and we call this fixed point a parabolic point. The parabolic points are necessarily in \(L(\Gamma)\). For a parabolic point \(z \in L(\Gamma)\), let \(k(z) \in \{1, \dots, d\}\) be the parabolic rank of \(z\) which is the rank of the largest free Abelian subgroup of the stabiliser of \(z\). The Hausdorff and box dimensions of \(L(\Gamma)\) are given by the Poincaré exponent \(\delta\) which is the critical exponent of the Poincaré series. Note that \[\frac{k(z)}{2} < \delta < d\] for all parabolic points \(z\), see [67]. The second inequality holds because we are in the geometrically finite case and we assumed the limit set was not the whole boundary. The natural measure supported on the limit set is the Patterson–Sullivan measure \(\mu\) and this is our measure of interest. This is a \(\delta\)-conformal measure of maximal dimension \(\dim_{\mathrm{H}}\mu = \delta = \dim_{\mathrm{H}}L(\Gamma)\) and is intimately connected to the action of \(\Gamma\) and the geometry of \(L(\Gamma)\).

Bourgain and Dyatlov [6] and later Li, Naud and Pan [8] proved that the Patterson–Sullivan measure enjoys polynomial Fourier decay in certain situations. In particular, it has positive Fourier dimension, but the precise value is non-quantitative and hard to pin down. As such, it is useful to provide concrete upper bounds and we do this here. We are able to use the local structure of the limit set near parabolic points to force mass into flat slabs, and use this to obtain quantitative flatness estimates.

One can see that the upper bound for the Fourier dimension in Theorem 43 (3) becomes very good if \(\delta\) is close to \[\max_z k(z)/2.\] This is a (strict) general lower bound for \(\delta\) but it is sharp and so by taking \(\delta\) arbitrarily close to \(\max_zk(z)/2>0\) we can exhibit examples where the Fourier dimension is arbitrarily small while the Hausdorff dimension remains bounded away from zero. Such examples are perhaps surprising given the work of Bourgain–Dyatlov [6] and Li–Naud–Pan [8] which established that the Fourier dimension is strictly positive in rather general situations.

The estimate in the case \(p=2\) in Theorem 43 (1) is always a non-trivial estimate whenever \(\delta<d\) in which case the lower bound for \(q\) is \(>2\). Moreover, it is always better than the bound \[q \geqslant\frac{2d}{\dim_{\mathrm{Fr}}\mu}\] coming from Theorem 18. This can be checked by using the formula for the Frostman dimension given in 15 above.

Theorem 43 gives simple to check conditions which ensure the Patterson–Sullivan measure \(\mu\) is not a Salem measure. We gave two conditions and neither is generally better than the other. As an example of where the first condition is better (and the second condition does not apply at all), we state a corollary in the case where there is a parabolic point of maximal rank.

Acknowledgements↩︎

I am grateful to Ana de Orellana for many useful conversations about Fourier restriction and Knapp examples, to Lijian Yang for helpful conversations about Fourier decay, and to Jeff Hicks for helpful conversations about differential geometry. I am also grateful to Amlan Banaji and Jonathan Hickman for helpful comments.

References↩︎