Fourier dimension of constant rank hypersurfaces

Junjie Zhu

Abstract.

Any hypersurface in $\mathbb{R}^{d+1}$ has a Hausdorff dimension of $d$ . However, the Fourier dimension depends on the finer geometric properties of the hypersurface. For example, the Fourier dimension of a hyperplane is 0, and the Fourier dimension of a hypersurface with non-vanishing Gaussian curvature is $d$ . Recently, Harris showed that the Euclidean light cone in $\mathbb{R}^{d+1}$ has a Fourier dimension of $d-1$ , which leads one to conjecture that the Fourier dimension of a hypersurface equals the number of non-vanishing principal curvatures. We prove this conjecture for all constant rank hypersurfaces. Our method involves substantial generalizations of Harris’s strategy.

Key words and phrases:

Fourier dimension, Hausdorff dimension, Constant rank hypersurfaces, Oscillatory integrals, Stationary phase

2020 Mathematics Subject Classification:

42B10, 42B20, 28A12, 53A07.

1. Introduction

The decay rate of the Fourier transforms of measures supported on manifolds has been of central interest in harmonic analysis. The Fourier transform of the normalized surface measure $\sigma$ on the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$ is given by

\widehat{\sigma}(\xi)=c_{d}|\xi|^{\frac{1-d}{2}}J_{\frac{d-1}{2}}(2\pi|\xi|),

where $J_{\frac{d-1}{2}}$ is the Bessel function of integral order $\frac{d-1}{2}$ , and $c_{d}$ is the normalizing constant [18]. It satisfies

(1)

|\widehat{\sigma}(\xi)|\leq C_{d}|\xi|^{-\frac{d}{2}}\text{ for a }C_{d}>0.

Describing sets in $\mathbb{R}^{d+1}$ , which may not be manifolds, using the Fourier decay of measures supported on them is one motivation for studying the Fourier dimension. Let $\mathcal{M}(A)$ be the set of measures $\mu$ supported on $A$ with finite total mass $\left\lVert\mu\right\rVert_{1}:=\mu(A)<\infty$ . The Fourier transform of a measure $\mu$ at $\xi\in\mathbb{R}^{n}$ is defined as $\widehat{\mu}(\xi):=\int e^{-2\pi ix\xi}d\mu(x)$ . The Fourier dimension for a Borel $A\subset\mathbb{R}^{d+1}$ is defined as

\dim_{F}(A):=\sup\{s\in[0,d+1]:\exists\mu\in\mathcal{M}(A),\sup_{\xi\in\mathbb{R}^{d+1}}|\xi|^{\frac{s}{2}}|\widehat{\mu}(\xi)|<\infty\}.

The notion of the Fourier dimension is ubiquitous in harmonic analysis and geometric measure theory. It provides a lower bound on the Hausdorff dimension, which is discussed further in Section 4.1. Studies of Fourier dimensions include probabilistic [17, 10, 1, 12] and deterministic constructions [11, 6] of Salem sets with equal Fourier and Hausdorff dimensions and properties satisfied by any Borel sets of large Fourier dimensions [13].

The objective of this paper is to study the case where $A$ is a constant rank hypersurface and compute its Fourier dimension.

1.1. Constant rank hypersurfaces

Let $M\subset\mathbb{R}^{d+1}$ be an orientable smooth hypersurface, which is a topological manifold of dimension $d$ with a normal direction $N:M\to\mathbb{S}^{d}$ . A way to describe $M$ is through notions of curvatures, defined through the eigenvalues of the Weingarten map [19]. Let $T_{p}M$ be the tangent space of $M$ at $p\in M$ . The Weingarten map $L_{p}:T_{p}M\to T_{p}M$ at $p\in M$ is the linear map $L_{p}(v)=-D_{v}N=-\frac{d}{dt}(N\circ\gamma)(0)$ , where $\gamma:I\to M$ is a curve with $\gamma(0)=p$ , $\gamma^{\prime}(0)=v$ . The principal curvatures of $M$ at $p$ are the eigenvalues of the map $L_{p}$ , and the Gaussian curvature of $M$ at $p$ is the product of the eigenvalues, which equals the determinant of $L_{p}$ . We note that Gaussian and principal curvatures at $p\in M$ are independent of the parametrization of $M$ and the choice of a basis for $T_{p}M$ . The Weingarten map is self-adjoint, so the eigenvalues of $L_{p}$ are real.

In this manuscript, we focus on hypersurfaces of constant rank $k$ where at all points $p\in M$ , $k$ principal curvatures are non-zero, and $d-k$ principal curvatures are zero. For example, in addition to cones and cylinders, tangent surfaces are hypersurfaces of constant rank $1$ in $\mathbb{R}^{3}$ . Two tangent surfaces are shown in Section 2. A hypersurface in higher dimensions not classified as a cone or a cylinder is

\left\{\left.\gamma(t)+\sum_{j=1}^{d-1}v_{j}\frac{d^{j}\gamma}{dt^{j}}(t)\right\rvert t,v_{j}\in\mathbb{R}\right\}\subset\mathbb{R}^{d+1},

where $\gamma(t)=(t,t^{2},\cdots,t^{d},t^{d+1})$ . It is of constant rank $1$ that generalizes tangent surfaces in $\mathbb{R}^{3}$ to higher dimensions. Another hypersurface of constant rank $2$ in $\mathbb{R}^{4}$ is

\left\{(t,\sin(t)e^{-s},\sin(2t)e^{-4s},\sin(3t)e^{-9s})+v(1,\cos(t)e^{-s},2\cos(2t)e^{-4s},3\cos(3t)e^{-9s})|t,s,v\in\mathbb{R}\right\}.

Some more examples are cylinders of cones $\{(kv,k,h)|k,h\in\mathbb{R},v\in S\}\subset\mathbb{R}^{d+1}$ , where $S$ is a hypersurface in $\mathbb{R}^{d-1}$ with non-vanishing Gaussian curvature.

Although surfaces of constant rank have been studied extensively in differential geometry [20], they are less examined in geometric measure theory, where geometric objects are analyzed via measures supported on them.

1.2. Main Result

This manuscript focuses on computing $\dim_{F}(M)$ , interpreting the properties of $M$ that $\dim_{F}(M)$ captures, and differentiating between topological, Hausdorff, and Fourier dimensions on $M$ . Since the Hausdorff dimension is preserved under diffeomorphisms, which are bi-Lipschitz, $\dim_{H}(M)=d$ [4, Proposition 3.1].

By (1), the unit sphere $\mathbb{S}^{d}$ has Fourier dimension $d$ . Hlawka [9] showed that hypersurfaces with non-vanishing Gaussian curvature also have a Fourier dimension of $d$ , which suggests that the Fourier dimension does not depend on the symmetry of the hypersurface. Harris [5] showed that the Euclidean light cone in $\mathbb{R}^{d+1}$ , which is a hypersurface with zero Gaussian curvature but of constant rank $d-1$ , has a Fourier dimension of $d-1$ . The proof relies on the rotational symmetry of the light cone. The author’s previous work [21] adapts the proof of [5] and shows that cones and cylinders of rank $d-1$ without rotational symmetry also have a Fourier dimension of $d-1$ . Table 1 summarizes these results.

Hypersurface $M$	Constant Rank	$\dim_{F}(M)$	Reference
non-vanishing Gaussian curvature ( $\mathbb{S}^{d}$ , etc)	$d$	$d$	[9]
Light cone $\{h(x,1)\|x\in\mathbb{S}^{d-1},h\in\mathbb{R}\}$	$d-1$	$d-1$	[5]
Generalized cone $\{h(x,1)\|x\in S,h\in\mathbb{R}\}$	$d-1$	$d-1$	[21]
$(d-1)$ -cylinder $S\times\mathbb{R}$	$d-1$	$d-1$	[21]
Hyperplane $\{x:x_{d+1}=0\}$	$0$	$0$

Table 1. Fourier dimension of some hypersurfaces, where

S\subset\mathbb{R}^{d}

is a hypersurface with non-zero Gaussian curvature.

Previous results suggest that the Fourier dimension of $M$ equals its rank. In this manuscript, we confirm that such a hypothesis holds

Theorem 1.1.

Let $M$ be a smooth hypersurface of constant rank $k$ in $\mathbb{R}^{d+1}$ , then $\dim_{F}(M)=k$ .

Remark.

a.

Theorem 1.1 shows a direct link between the principal curvatures and the Fourier dimension for constant rank hypersurfaces. We can view the Fourier dimension as a generalization of the number of non-zero principal curvatures defined on hypersurfaces. It is a generalization of the results in [5, 21]. It includes all hypersurfaces of constant rank $k$ for $k<d-1$ and hypersurfaces of constant rank $d-1$ that are neither cones nor cylinders.
b.

Previously known bounds on the Fourier dimension of a constant rank $k$ hypersurface $M\subset\mathbb{R}^{d+1}$ are

$k\leq\dim_{F}(M)\leq k\left(\frac{d+1}{k+1}\right).$

The lower bound from [14] arises from the Fourier transform of measures supported on the hypersurface induced by the Lebesgue measure. The upper bound is obtained from the study of the $p$ -thin problem [7]. If $\mu\in\mathcal{M}(M)$ satisfies $|\widehat{\mu}(\xi)|\leq C(1+|\xi|)^{-\frac{\alpha}{2}}$ for $C,\alpha>0$ , $\widehat{\mu}\in L^{p}(\mathbb{R}^{d+1})$ for all $p>\frac{2(d+1)}{\alpha}$ . Such $p$ must satisfy $p>\frac{2(k+1)}{k}$ , so $\alpha\leq k\left(\frac{d+1}{k+1}\right)$ .

1.3. Overview of the proof

To establish Theorem 1.1, we need to show that the Fourier dimension of a hypersurface $M$ of constant rank $k$ is bounded below and above by $k$ . For the upper bound (Proposition 1.2), the main strategy adapted from [5] is to create a new measure by averaging a series of push-forward of an old measure supported on $M$ .

An adaptation of the methodology from [5] is needed. In previous works [5, 21], it was believed that the specific maps used to push-forward measures are required to map $M$ to itself, so the new measure is still supported on the hypersurface. While such assumptions pose no issue for cones and cylinders, similar maps may not exist for more general hypersurfaces for us to apply the same machinery.

Fortunately, we note that the new measure does not have to be supported on the original $M$ . We propose new maps that push forward an old measure to a series of similar hypersurfaces, and we still create a new measure as the average of the series of measures. The proof in Section 3 shows that such a new measure may not be supported on $M$ , but it has a Fourier decay exponent in one direction that is not less than that of the old measure. Moreover, the Fourier transform of the new measure is related to the second fundamental form of the hypersurface $M$ , which prevents the Fourier decay exponent of the new measure from being greater than $k/2$ .

1.4. Proof of the main result and organization of the manuscript

Theorem 1.1 follows from Theorem A and Proposition 1.2.

Theorem A ([14]).

Suppose that at least $k$ of the principal curvatures are not zero at all points in $M$ . Then, there exists a measure $\mu\in\mathcal{M}(M)$ such that $\sup_{\xi\in\mathbb{R}^{d+1}}|\xi|^{\frac{k}{2}}|\widehat{\mu}(\xi)|<\infty$ .

Therefore, if $M$ is a constant rank $k$ hypersurface, $\dim_{F}(M)\geq k$ . It remains to prove the following.

Proposition 1.2.

$\dim_{F}(M)\leq k$ .

We present examples of Proposition 1.2 in Section 2 and prove Proposition 1.2 in Section 3.

1.5. Notations

For two functions $f,g:D\to\mathbb{R}_{\geq 0}$ with a domain $D$ , we write $f\lesssim g$ to denote that there exists a $c>0$ , such that for all $x\in D$ , $f(x)\leq cg(x)$ .

For $\phi:\mathbb{R}^{n}\to\mathbb{R}^{n}$ , we denote the Jacobian of $\phi$ as ${\bf J}\phi$ .

If $(X,\mathcal{X})$ and $(Y,\mathcal{Y})$ are measurable spaces and the mapping $F:X\to Y$ is $(\mathcal{X},\mathcal{Y})$ -measurable, the push-forward map $F^{\#}$ is defined to be $F^{\#}\mu(U)=\mu(F^{-1}(U))$ for any set $U\in\mathcal{Y}$ and any measure $\mu$ on $(X,\mathcal{X})$ .

On $(X,\mathcal{X})$ , if $f$ is a measurable function and $\mu$ is a measure, we denote $\langle f,\mu\rangle=\int_{X}fd\mu$ .

1.6. Acknowledgments

I thank Malabika Pramanik and Jialing Zhang for their discussions on this project. I thank Yuveshen Moorogen for editing suggestions for an earlier version of the manuscript.

2. Upper bounds on the Fourier dimension: Examples

In this section, we demonstrate the process of showing that the Fourier dimensions of two tangent surfaces, which are hypersurfaces of constant rank $1$ in $\mathbb{R}^{3}$ different from generalized cones and cylinders, are at most 1. We note that the normal vectors used in the section are not unit vectors, but the conclusion is not affected.

2.1. Tangent surface generated by a Helix

Let $\gamma:\mathbb{R}\to\mathbb{R}^{3}$ , $\gamma(t)=(t,t^{2},t^{3})$ , and

(2)

S=\{\Phi(t,v):=\gamma(t)+v\gamma^{\prime}(t)|t\in\mathbb{R},v\neq 0\}

be the tangent surface generated by $\gamma$ , where $\gamma^{\prime}(t)=(1,2t,3t^{2})$ . $\gamma^{\prime\prime}(t)=(0,2,6t)$ , and a normal vector of $S$ at a point $\Phi(t,v)$ is

(3)

\overrightarrow{n}(t):=(3t^{2},-3t,1).

The tangent surface $S$ is of rank 1 since the second fundamental form at $\Phi(t,v)$ is

\begin{pmatrix}\Phi_{tt}\cdot\overrightarrow{n}&\Phi_{tv}\cdot\overrightarrow{n}\\ \Phi_{vt}\cdot\overrightarrow{n}&\Phi_{vv}\cdot\overrightarrow{n}\end{pmatrix}=\begin{pmatrix}6v&0\\ 0&0\end{pmatrix}

has rank $1$ when $v\neq 0$ .

Refer to caption — Figure 1. A 3D plot of $S$ from Matlab.

Proposition 2.1.

$\dim_{F}(S)\leq 1$ .

Proof.

Let $\mu\in\mathcal{M}(S)$ . Suppose that there exists $\beta>0$ , such that $|\widehat{\mu}(\xi)|\leq C|\xi|^{-\frac{\beta}{2}}$ for $\xi\in\mathbb{R}^{3}$ , $C>0$ . By Lemma 4.2, we may assume that $S=\{\Phi(t,v)||t|\leq c,v\in[a,b]\}$ for $0<a<b$ , $c>0$ .

We construct new measures. Let the reparametrization $T_{s}:S\to S$ be

(4)

T_{s}(\Phi(t,v))=\Phi(t-s,v).

Figure 2. Illustration of the re-parametrization by

T_{s}

S

Let $\mu_{s}\in\mathcal{M}(S)$ be

(5)

\mu_{s}=T_{s}^{\#}\mu.

Let $\psi\in C_{0}^{\infty}(\mathbb{R})$ be non-negative bump function with $\psi(s)=1$ when $|s|\leq c$ , $\psi(s)=0$ when $|s|\geq 2c$ . Then we let the average measure $\nu$ be defined as

(6)

\int fd\nu=\int\langle f,\mu_{s}\rangle\psi(s)ds

for non-negative Borel functions $f$ .

Now, we examine $\widehat{\nu}$ in the $e_{3}$ direction, where $e_{3}$ is also a normal vector at $\gamma(0,v)$ . Proposition 2.1 is a consequence of the following two lemmas.

Lemma 2.2.

For $\rho>0$ , $|\widehat{\nu}(\rho e_{3})|\lesssim\rho^{-\frac{\beta}{2}}$ .

Lemma 2.3.

There exists $\rho_{0}>0$ , such that for $\rho>\rho_{0}$ , $|\widehat{\nu}(\rho e_{3})|\gtrsim\rho^{-\frac{1}{2}}$ . ∎

Proof of Lemma 2.2.

First, we show that for each $s\in\mathbb{R}$ , $|\widehat{\mu_{s}}(\rho e_{3})|\lesssim\rho^{-\frac{\beta}{2}}$ . By unwinding the definitions of push-forward measures, Fourier transform, and integration on the manifold, we have

(7)	$\displaystyle\widehat{\mu_{s}}(\rho e_{3})$	$\displaystyle=\int e^{-2\pi ix\cdot\rho e_{3}}dT_{s}^{\#}\mu(x)$	by (5)
		$\displaystyle=\int e^{-2\pi i\rho T_{s}(\Phi(t,v))\cdot e_{3}}d\mu\circ\Phi(t,v)$
		$\displaystyle=\int e^{-2\pi i\rho\Phi(t-s,v)\cdot e_{3}}d\mu\circ\Phi(t,v)$	$\displaystyle\text{ by (\ref{eq31_TS})}.$

From (2) and (3),

(8)

\Phi(t-s,v)\cdot e_{3}=\Phi(t,v)\cdot\overrightarrow{n}(s)-s^{3}.

Then, (7) becomes

((7) c.)	$\displaystyle\widehat{\mu_{s}}(\rho e_{3})$	$\displaystyle=\int e^{-2\pi i\rho\Phi(t-s,v)\cdot e_{3}}d\mu\circ\Phi(t,v)$
		$\displaystyle=\int e^{-2\pi i\rho[\Phi(t,v)\cdot\overrightarrow{n}(s)-s^{3}]}d\mu\circ\Phi(t,v)$	by (8)
		$\displaystyle=e^{\pi i\rho s^{3}}\widehat{\mu}(\rho\overrightarrow{n}(s)).$

Since $|\overrightarrow{n}(s)|\geq 1$ , $|\widehat{\mu_{s}}(\rho e_{3})|=|\widehat{\mu}(\rho\overrightarrow{n}(s))|\leq C|\rho\overrightarrow{n}(s)|^{-\frac{\beta}{2}}\leq C\rho^{-\frac{\beta}{2}}.$ Then,

	$\displaystyle\|\widehat{\nu}(\rho e_{3})\|\leq$	$\displaystyle\int\|\widehat{\mu_{s}}(\rho\overrightarrow{n}(s))\|\psi(s)ds$	by (6)
	$\displaystyle\leq$	$\displaystyle C\left\lVert\psi\right\rVert_{L^{1}}\rho^{-\frac{\beta}{2}}.$

Proof of Lemma 2.3.

An expression of $\widehat{\nu}(\rho e_{3})$ is given by

(9)		$\displaystyle\widehat{\nu}(\rho e_{3})$	$\displaystyle=\int\int e^{-2\pi i\rho\Phi(t-s,v)\cdot e_{3}}d\mu\circ\Phi(t,v)\psi(s)ds$	by (6)
(9)			$\displaystyle=\int\int e^{-2\pi i\rho\Phi(t-s,v)\cdot e_{3}}\psi(s)dsd\mu\circ\Phi(t,v).$

We apply the stationary phase method to study the inner integral

I(\rho;t,v):=\int e^{-2\pi i\rho\Phi(t-s,v)\cdot e_{3}}\psi(s)ds.

Fixing $t,v$ , we define the phase function $\phi$ of $I(\rho;t,v)$ as

\phi(s):=\Phi(t-s,v)\cdot e_{3}=(t-s)^{3}+3v(t-s)^{2}\text{ from }(\ref{eq_helix_Phie3}).

Its derivatives are

\phi^{\prime}(s)=-3(t-s)^{2}-6v(t-s),

and

\phi^{\prime\prime}(s)=6(t-s)+6v.

The solutions to $\phi^{\prime}(s)=0$ are

s=t,t+2v.

Since $v\in[a,b]$ , if $-2c+2a>2c$ or $2c<a$ , the only critical point in $\text{spt }\psi$ is $s=t$ . Note that

\phi(t)=0,\phi^{\prime}(t)=0,\phi^{\prime\prime}(t)=6v.

Then, by the stationary phase (part b, Theorem 4.6), there exists $D_{0}>0$ independent of $\rho$ , $t$ , and $v$ , such that when $\rho v$ is sufficiently large,

\left|I(\rho;t,v)-\left(\frac{-i}{6\rho v}\right)^{\frac{1}{2}}\psi(t)\right|\leq D_{0}(\rho v)^{-1}.

Since $\psi(t)=1$ for $|t|\leq c$ , from (9),

\begin{split}\left|\widehat{\nu}(\rho e_{3})-\int\left(\frac{-i}{6\rho v}\right)^{\frac{1}{2}}d\mu\circ\Phi(t,v)\right|&\leq\int D_{0}(\rho v)^{-1}d\mu\circ\Phi(t,v).\\ \end{split}

As $v\in[a,b]$ ,

	$\displaystyle\left\|\int\left(\frac{-i}{6\rho v}\right)^{\frac{1}{2}}d\mu\circ\Phi(t,v)\right\|\geq$	$\displaystyle 6^{-\frac{1}{2}}b^{-\frac{1}{2}}\rho^{-\frac{1}{2}},$
	$\displaystyle\left\|\int D_{0}(\rho v)^{-1}d\mu\circ\Phi(t,v)\right\|\leq$	$\displaystyle D_{0}a^{-1}\rho^{-1}.$

Therefore, $|\widehat{\nu}(\rho e_{3})|$ is bounded below by $4^{-1}b^{-\frac{1}{2}}\rho^{-\frac{1}{2}}$ for $\rho$ sufficiently large. ∎

2.2. A tangent surface generated by a perturbed helix

In this subsection, we examine the tangent surface generated by the curve $\alpha(t)=(t,t^{2}+t^{4},t^{3})$ .

Proposition 2.4.

Let $I=[-2c,2c]$ for $c>0$ , and let $\alpha:I\to\mathbb{R}^{3}$ be defined above. Let

S:=\{\alpha(t)+v\alpha^{\prime}(t)|t\in I,a<v<b\}

be the tangent surface generated by $\alpha$ , where $0<a<b$ . Then, $\dim_{F}(S)\leq 1$ .

We denote $\Phi:I\times(a,b)\to\mathbb{R}^{3}$ be $\Phi(t,v)=\alpha(t)+v\alpha^{\prime}(t)$ . Proposition 2.4 holds if the following lemma as an analog of (8) holds.

Lemma 2.5.

For $s\in I$ , there exists $T_{s}:S\to\mathbb{R}^{3}$ , such that

(10)

T_{s}(\alpha(t)+v\alpha^{\prime}(t))\cdot e_{3}=(\alpha(t)+v\alpha^{\prime}(t)-\alpha(s))\cdot\overrightarrow{n}(s),

where

(11)

\overrightarrow{n}(s)=(-3s^{2}(2s^{2}-1),-3s,6s^{2}+1)

is a normal vector of $S$ at $\Phi(s,v)$ with $\overrightarrow{n}(0)=e_{3}$ .

Proof of Proposition 2.4 assuming Lemma 2.5.

Let $\mu\in\mathcal{M}(S)$ , and suppose that there exists $\beta>0$ , such that $|\widehat{\mu}(\xi)|\leq C|\xi|^{-\frac{\beta}{2}}$ for $\xi\in\mathbb{R}^{3}$ , $C>0$ .

We construct new measures. Let $\mu_{s}\in\mathcal{M}(\mathbb{R}^{3})$ be

(12)

\mu_{s}=T_{s}^{\#}\mu.

Let $\psi\in C_{0}^{\infty}(\mathbb{R})$ be a non-negative bump function with $\psi(s)=1$ when $|s|\leq c$ , $\psi(s)=0$ when $|s|\geq 2c$ . Then we let the average measure $\nu$ be defined as

(13)

\int fd\nu=\int\langle f,\mu_{s}\rangle\psi(s)ds

for non-negative Borel functions $f$ . Proposition 2.4 is a consequence of the following two lemmas.

Lemma 2.6.

For $\rho>0$ , $|\widehat{\nu}(\rho e_{3})|\lesssim\rho^{-\frac{\beta}{2}}$ .

Lemma 2.7.

There exists $\rho_{0}>0$ , such that for $\rho>\rho_{0}$ , $|\widehat{\nu}(\rho e_{3})|\gtrsim\rho^{-\frac{1}{2}}$ . ∎

Compared with previous methods, the computations in Lemmas 2.6 and 2.7 do not require $\text{spt }\nu\subset S$ .

Proof of Lemma 2.6.

First, we show that for each $s\in I$ , $|\widehat{\mu_{s}}(\rho e_{3})|\lesssim\rho^{-\frac{\beta}{2}}$ . By unwinding the definitions of the push-forward measure, Fourier transform, and integration on the manifold, we have

(14)	$\displaystyle\widehat{\mu_{s}}(\rho e_{3})$	$\displaystyle=\int e^{-2\pi iy\cdot\rho e_{3}}dT_{s}^{\#}\mu(y)$	by (12)
		$\displaystyle=\int e^{-2\pi i\rho T_{s}(\Phi(t,v))\cdot e_{3}}d\mu\circ\Phi(t,v)$
		$\displaystyle=\int e^{-2\pi i\rho[(\alpha(t)+v\alpha^{\prime}(t))\cdot\overrightarrow{n}(s)-\alpha(s)\cdot\overrightarrow{n}(s)]}d\mu\circ\Phi(t,v)$	by (10)
		$\displaystyle=e^{2\pi i\rho\alpha(s)\cdot\overrightarrow{n}(s)}\widehat{\mu}(\rho\overrightarrow{n}(s)).$

	$\displaystyle\|\widehat{\nu}(\rho e_{3})\|\leq$	$\displaystyle\int\|\widehat{\mu_{s}}(\rho e_{3})\|\psi(s)ds$	by (13)
	$\displaystyle\leq$	$\displaystyle C\left\lVert\psi\right\rVert_{L^{1}}\rho^{-\frac{\beta}{2}}.$

Proof of Lemma 2.7.

An expression of $\widehat{\nu}(\rho e_{3})$ is given by

(15)		$\displaystyle\widehat{\nu}(\rho e_{3})$	$\displaystyle=\int\int e^{-2\pi i\rho T_{s}(\Phi(t,v))\cdot e_{3}}d\mu\circ\Phi(t,v)\psi(s)ds$	by (13)
(15)			$\displaystyle=\int\int e^{-2\pi i\rho T_{s}(\Phi(t,v))\cdot e_{3}}\psi(s)dsd\mu\circ\Phi(t,v).$

We apply the stationary phase method to study the inner integral

I(\rho;t,v):=\int e^{-2\pi i\rho T_{s}(\Phi(t,v))\cdot e_{3}}\psi(s)ds.

Fixing $t,v$ , we define the phase function $\phi$ of $I(\rho;t,v)$ as

	$\displaystyle\phi(s)$	$\displaystyle:=T_{s}(\Phi(t,v))\cdot e_{3}=(\alpha(t)+v\alpha^{\prime}(t))\cdot\overrightarrow{n}(s)-\alpha(s)\cdot\overrightarrow{n}(s)$	$\displaystyle\text{ by }(\ref{p_phelix_ex})$
		$\displaystyle=[(t-s,t^{2}+t^{4}-s^{2}-s^{4},t^{3}-s^{3})+v(1,2t+4t^{3},3t^{2})]\cdot(-6s^{4}+3s^{2},-3s,6s^{2}+1)$

Its first derivative is

\begin{split}\phi^{\prime}(s)=&-3(t-s)[(t-s)^{3}+4v(t-s)^{2}+(1-6s^{2})(t-s)+2v(1-6s^{2})].\end{split}

One critical point is $s=t$ . If $c$ is sufficiently small, the only critical point in $\text{spt }\psi$ is $s=t$ . Note that

\phi(t)=0,\phi^{\prime}(t)=0,\phi^{\prime\prime}(t)=v(6-36t^{2}).

Then, by the stationary phase (part b of Theorem 4.6),

\left|I(\rho;t,v)-\left(\frac{-i}{\rho v(6-36t^{2})}\right)^{\frac{1}{2}}\psi(t)\right|\leq D_{1}(\rho v)^{-\frac{3}{2}}

for a $D_{1}>0$ independent of $\rho$ , $t$ , and $v$ . Since $\psi(t)=1$ for $|t|\leq c$ , continuing with (15),

\begin{split}\left|\widehat{\nu}(\rho e_{3})-\int\left(\frac{-i}{\rho v(6-36t^{2})}\right)^{\frac{1}{2}}d\mu\circ\Phi(t,v)\right|&\leq\int D_{1}(\rho v)^{-1}d\mu\circ\Phi(t,v)\\ \end{split}

As $v\in[a,b]$ ,

	$\displaystyle\left\|\int\left(\frac{-i}{\rho v(6-36t^{2})}\right)^{\frac{1}{2}}d\mu\circ\Phi(t,v)\right\|\geq$	$\displaystyle 6^{-\frac{1}{2}}b^{-\frac{1}{2}}\rho^{-\frac{1}{2}},$
	$\displaystyle\left\|\int D_{1}(\rho v)^{-1}d\mu\circ\Phi(t,v)\right\|\leq$	$\displaystyle D_{1}a^{-1}\rho^{-1}.$

Therefore, $|\widehat{\nu}(\rho e_{3})|$ is bounded below by $4^{-1}b^{-\frac{1}{2}}\rho^{-\frac{1}{2}}$ for $\rho$ sufficiently large. ∎

Now, we show the existence of $T_{s}$ for the perturbed helix.

Proof of Lemma 2.5.

To satisfy (10), a possible $T_{s}$ fixes the first two coordinates of $\alpha(t)+v\alpha^{\prime}(t)$ and updates the third coordinate to the right-hand side of (10). ∎

Remark.

The tangent surface generated by the curve $\gamma(t)=(t,t^{2},t^{3})$ presented in Section 2.1 is one case where $T_{s}$ satisfies (10) and $T_{s}(S)\subset S$ . For the example in this section, it is not possible to ensure that $T_{s}(S)\subset S$ and $\mu_{s},\nu$ are still supported on $S$ , even if we only require $\overrightarrow{n}(s)$ to be any normal vector of $S$ at $\Phi(s,v)$ and have a norm comparable to 1.

One interpretation is that $T_{s}$ defined in the proof maps $S\to S_{s}$ , where $S_{s}$ is the tangent surface generated by the curve

\alpha_{s}(t)=\langle t,t^{2}+t^{4},6s^{3}(s-1)(t-s)+(1-6s^{2})(t-s)^{3}-3s(t-s)^{4}\rangle,

and

T_{s}(\alpha(t)+v\alpha^{\prime}(t))=\alpha_{s}(t)+v\alpha_{s}^{\prime}(t).

Figure 3. Illustration of the map

T_{s}:S\to S_{s}

3. Proof of Proposition 1.2

We use terminologies commonly used in describing hypersurfaces of constant rank.

Definition 3.1.

$M$ is a $(d-k)$ -ruled hypersurface if near $p\in M$ , a coordinate chart is given by $\Phi:U\times V\subset\mathbb{R}^{k}\times\mathbb{R}^{d-k}\to S$ as

(16)

\Phi(u,v)=\alpha(u)+\sum_{l=1}^{d-k}v_{l}w_{l}(u),

with $\alpha,w_{1},\ldots,w_{d-k}:\mathbb{R}^{k}\to\mathbb{R}^{d+1}$ . A ruling is $\{\Phi(u,v)|v\in V\}$ for a fixed $u$ .

Let $N(u,v)$ be the normal vector of $M$ at $\Phi(u,v)$ . In the proof of Proposition 1.2, we rely on the following parametrization on hypersurfaces with constant rank $k$ .

Lemma B.

[8, Lemma 3.1] [20] The following statements are equivalent.

(1)

$M$ is a smooth hypersurface in $\mathbb{R}^{d+1}$ of constant rank $k$ .
(2)

$M$ is $(d-k)$ -ruled, and the normal vectors along the rulings are constant.

We deduce the analytic properties of constant rank hypersurfaces that will be applied throughout this section.

The normal vectors along the rulings being constant is equivalent to

(17)

N(u,v)=N(u,v^{\prime})

for $v\neq v^{\prime}$ . We may denote $N(u)=N(u,v)$ for a $v\in V$ . Since $N(u,v)$ is the normal vector at $p=\Phi(u,v)$ , it is perpendicular to all vectors in $T_{p}M$ . Therefore,

(18)

\begin{split}\left\langle\frac{\partial\Phi}{\partial u_{j}}(u,v),N(u,v)\right\rangle&=\left\langle\frac{\partial\alpha}{\partial u_{j}}(u)+\sum_{l=1}^{d-k}v_{l}\frac{\partial w_{l}}{\partial u_{j}}(u),N(u,v)\right\rangle=0,\\ \left\langle\frac{\partial\Phi}{\partial v_{l}}(u,v),N(u,v)\right\rangle&=\left\langle w_{l}(u),N(u,v)\right\rangle=0.\end{split}

From the first equation of (18), for a fixed $u$ , since $N(u,v)=N(u,v^{\prime})$ when $v\neq v^{\prime}$ , for all $l$ and $j$ ,

\left\langle\frac{\partial w_{l}}{\partial u_{j}}(u),N(u,v)\right\rangle=0.

By differentiating the second equation in (18) with respect to $u_{j}$ , we obtain

(19)

\left\langle w_{l}(u),\frac{\partial N}{\partial u_{j}}(u,v)\right\rangle=0.

The second fundamental form of $M$ at $p$ has rank $k$ . Since

\frac{\partial^{2}\Phi}{\partial v_{l}\partial v_{l^{\prime}}}(u,v)=0,

and

\left\langle\frac{\partial^{2}\Phi}{\partial u_{j}\partial v_{l}}(u,v),N(u,v)\right\rangle=\left\langle\frac{\partial w_{l}}{\partial u_{j}}(u),N(u,v)\right\rangle=0,

the top-left $k\times k$ submatrix of the second fundamental form

\left(\frac{\partial^{2}\Phi}{\partial u_{j}\partial u_{j^{\prime}}}(u,v)\cdot N(u,v)\right)

has full rank $k$ .

By rotation and translation, we may assume that $U$ is a neighborhood of $0$ , $N(0)=e_{d+1}\in\mathbb{R}^{d+1}$ , where $\{e_{1},\cdots,e_{d+1}\}$ is the standard basis of $\mathbb{R}^{d+1}$ .

Remark.

a.

A $(d-k)$ -ruled hypersurface does not have a constant rank of $k$ if the normal vectors are not constant along the rulings, and an $m$ -ruled hypersurface can also be viewed as an $(m-1)$ -ruled hypersurface. For example, in $\mathbb{R}^{3}$ , the hyperboloid given by $x^{2}+y^{2}=z^{2}+1$ is $1$ -ruled, but it is a hypersurface with non-vanishing Gaussian curvature. [2, Sec. 3-5] The normal vectors of the hyperboloid are not constant along the rulings.
b.

The proof of Theorem A uses Morse’s Lemma (Lemma 4.3) to parametrize a hypersurface $M$ . The author’s previous work [21] partly uses Morse’s Lemma to parametrize cones and cylinders. A version of Proposition 1.2 could also be proven via Morse’s Lemma; however, the conclusion would be the same as the Fourier dimension is independent of the parametrizations.
c.

The hypersurface $M$ is cylindrical if $w_{l}(u_{1})=w_{l}(u_{2})$ for all $u_{1},u_{2}\in U$ , $1\leq l\leq d-k$ . There is a simpler proof in Lemma 4.1 for a cylindrical hypersurface since it can be written as $S\times\mathbb{R}^{d-k}$ for a hypersurface $S\subset\mathbb{R}^{k+1}$ with non-vanishing Gaussian curvature, and it follows that $\dim_{F}(M)=\dim_{F}(S)=k$ .

3.1. The Main Proof

Proposition 1.2 holds if the following lemma holds.

Lemma 3.2.

For $s\in U$ , there exists $T_{s}:M\to\mathbb{R}^{d+1}$ , such that

(20)

T_{s}(\Phi(u,v))\cdot e_{d+1}=(\Phi(u,v)-\alpha(s))\cdot N(s)

for the unit normal vector $N(s)$ .

Proof of Proposition 1.2 assuming Lemma 3.2.

Let $\mu\in\mathcal{M}(M)$ , and suppose that there exists $\beta>0$ , such that $|\widehat{\mu}(\xi)|\leq C|\xi|^{-\frac{\beta}{2}}$ for $\xi\in\mathbb{R}^{d+1}$ , $C>0$ . By Lemma 4.2, we may assume $\mu\in\mathcal{M}(\Phi(\frac{1}{2}U,V))$ .

We construct new measures. Let $\mu_{s}\in\mathcal{M}(\mathbb{R}^{d+1})$ be

(21)

\mu_{s}=T_{s}^{\#}\mu.

Let $\psi\in C_{0}^{\infty}(\mathbb{R}^{k})$ be non-negative bump function with $\psi(s)=1$ when $s\in\frac{1}{2}U$ , $\psi(s)=0$ when $s\in(\frac{3}{4}U)^{c}$ . Then, we define the average measure $\nu\in\mathcal{M}(\mathbb{R}^{d+1})$ as

(22)

\int fd\nu=\int\langle f,\mu_{s}\rangle\psi(s)ds

for non-negative Borel functions $f$ via the Riesz representation theorem for positive linear functionals [16]. Proposition 1.2 is a consequence of the following two lemmas.

Lemma 3.3.

For $\rho>0$ , $|\widehat{\nu}(\rho e_{d+1})|\lesssim\rho^{-\frac{\beta}{2}}$ .

Lemma 3.4.

There exists $\rho_{0}>0$ , such that for $\rho>\rho_{0}$ , $|\widehat{\nu}(\rho e_{d+1})|\gtrsim\rho^{-\frac{k}{2}}$ . ∎

Proof of Lemma 3.3.

First, we show that for each $s\in U$ , $|\widehat{\mu_{s}}(\rho e_{d+1})|\lesssim\rho^{-\frac{\beta}{2}}$ . By unwinding the definitions of the push-forward measure, Fourier transform, and integration on the manifold, we have

(23)	$\displaystyle\widehat{\mu_{s}}(\rho e_{d+1})$	$\displaystyle=\int e^{-2\pi iy\cdot\rho e_{d+1}}dT_{s}^{\#}\mu(y)$	by (21)
		$\displaystyle=\int e^{-2\pi i\rho T_{s}(\Phi(u,v))\cdot e_{d+1}}d\mu\circ\Phi(u,v)$
		$\displaystyle=\int e^{-2\pi i\rho[\Phi(u,v)\cdot N(s)-\alpha(s)\cdot N(s)]}d\mu\circ\Phi(u,v)$	by (20)
		$\displaystyle=e^{2\pi i\rho\alpha(s)\cdot N(s)}\widehat{\mu}(\rho N(s)).$

Since $|N(s)|=1$ , $|\widehat{\mu_{s}}(\rho e_{d+1})|=|\widehat{\mu}(\rho N(s))|\leq C\rho^{-\frac{\beta}{2}}.$ Then

	$\displaystyle\|\widehat{\nu}(\rho e_{d+1})\|\leq$	$\displaystyle\int\|\widehat{\mu_{s}}(\rho N(s))\|\psi(s)ds$	by (22)
	$\displaystyle\leq$	$\displaystyle C\left\lVert\psi\right\rVert_{L^{1}}\rho^{-\frac{\beta}{2}}.$

Proof of Lemma 3.4.

An expression of $\widehat{\nu}(\rho e_{d+1})$ is given by

(24)		$\displaystyle\widehat{\nu}(\rho e_{d+1})$	$\displaystyle=\int\int e^{-2\pi i\rho T_{s}(\Phi(u,v))\cdot e_{d+1}}d\mu\circ\Phi(u,v)\psi(s)ds$	by (22)
(24)			$\displaystyle=\int\int e^{-2\pi i\rho T_{s}(\Phi(u,v))\cdot e_{d+1}}\psi(s)dsd\mu\circ\Phi(u,v).$

We apply the stationary phase method to study the inner integral

I(\rho;u,v):=\int e^{-2\pi i\rho T_{s}(\Phi(u,v))\cdot e_{d+1}}\psi(s)ds.

Fixing $u,v$ , we define the phase function $\phi$ of $I(\rho;t,v)$ as

\phi(s):=T_{s}(\Phi(u,v))\cdot e_{d+1}=\Phi(u,v)\cdot N(s)-\alpha(s)\cdot N(s)\text{ by }(\ref{e621}).

Two claims about the critical point of $\phi$ are as follows:

•

One critical point is $s=u$ : by (19),

\begin{split}\frac{\partial\phi}{\partial s_{j}}(u)=&(\Phi(u,v)-\alpha(u))\cdot\frac{\partial N}{\partial u_{j}}(u)\\ =&\left(\alpha(u)+\sum_{l=1}^{d-k}v_{l}w_{l}(u)-\alpha(u)\right)\cdot\frac{\partial N}{\partial u_{j}}(u)\\ =&\sum_{l=1}^{d-k}v_{l}w_{l}(u)\cdot\frac{\partial N}{\partial u_{j}}(u)=0.\end{split}

•

If $U$ is sufficiently small, no other critical point exists: we will show that the Hessian has full rank, so the critical points of the function $\phi$ are isolated. By differentiating (19) with respect to $u_{j^{\prime}}$ , we obtain

\left\langle\frac{\partial w_{l}}{\partial u_{j^{\prime}}}(u),\frac{\partial N}{\partial u_{j}}(u)\right\rangle+\left\langle w_{l}(u),\frac{\partial^{2}N}{\partial u_{j}\partial u_{j}^{\prime}}(u)\right\rangle=0

Then,

\begin{split}\frac{\partial^{2}\phi}{\partial s_{j}\partial s_{j^{\prime}}}(u)=&(\Phi(u,v)-\alpha(u))\cdot\frac{\partial^{2}N}{\partial u_{j}\partial u_{j^{\prime}}}(u)-\frac{\partial\alpha}{\partial u_{j^{\prime}}}(u)\cdot\frac{\partial N}{\partial u_{j}}(u)\\ =&-\frac{\partial\Phi}{\partial u_{j}^{\prime}}(u,v)\cdot\frac{\partial N}{\partial u_{j}}(u).\end{split}

By part b) of the discussion following Lemma B, the matrix $\left(\frac{\partial^{2}\Phi}{\partial u_{j}\partial u_{j^{\prime}}}(u,v)\cdot N(u)\right)$ has full rank $k$ . Therefore, $\Delta_{s}\phi(u)$ has full rank $k$ .

Then, by the stationary phase (part b of Theorem 4.6),

\left|I(\rho;u,v)-\rho^{-\frac{k}{2}}(-1)^{k}i^{\frac{k}{2}}\left(\det\left(\frac{\partial^{2}\Phi}{\partial u_{j}\partial u_{j^{\prime}}}(u,v)\cdot N(u)\right)\right)^{-\frac{1}{2}}\psi(u)\right|\leq D^{\prime}\rho^{-\frac{k+1}{2}}

for a $D^{\prime}>0$ independent of $\rho$ , $u$ , and $v$ . Since $\psi(u)=1$ for $u\in\frac{1}{2}U$ , continuing from (24),

\begin{split}&\left|\widehat{\nu}(\rho e_{d+1})-\int\rho^{-\frac{k}{2}}(-1)^{k}i^{\frac{k}{2}}\left(\det\left(\frac{\partial^{2}\Phi}{\partial u_{j}\partial u_{j^{\prime}}}(u,v)\cdot N(u)\right)\right)^{-\frac{1}{2}}d\mu\circ\Phi(u,v)\right|\\ \leq&\int D^{\prime}\rho^{-\frac{k+1}{2}}d\mu\circ\Phi(u,v).\\ \end{split}

Since

\det\left(\frac{\partial^{2}\Phi}{\partial u_{j}\partial u_{j^{\prime}}}(u,v)\cdot N(u)\right)\leq D

for a $D>0$ ,

	$\displaystyle\left\|\int\rho^{-\frac{k}{2}}(-1)^{k}i^{\frac{k}{2}}\left(\det\left(\frac{\partial^{2}\Phi}{\partial u_{j}\partial u_{j^{\prime}}}(u,v)\cdot N(u)\right)\right)^{-\frac{1}{2}}d\mu\circ\Phi(u,v)\right\|\geq$	$\displaystyle D^{-\frac{1}{2}}\rho^{-\frac{k}{2}},$
	$\displaystyle\left\|\int D^{\prime}\rho^{-\frac{k+1}{2}}d\mu\circ\Phi(u,v)\right\|\leq$	$\displaystyle D^{\prime}\rho^{-\frac{k+1}{2}}.$

Therefore, $|\widehat{\nu}(\rho e_{d+1})|$ is bounded below by $2^{-1}D^{-\frac{1}{2}}\rho^{-\frac{k}{2}}$ for $\rho$ sufficiently large. ∎

3.2. Existence of $T_{s}$

Proof of Lemma 3.2.

One choice of $T_{s}$ fixes the first $n$ coordinates of $\Phi(u,v)$ and changes the last coordinate to

(\Phi(u,v)-\alpha(s))\cdot N(s)=(\alpha(u)-\alpha(s))\cdot N(s)+\sum_{l=1}^{d-k}v_{l}w_{l}(u)\cdot N(s).

Alternatively,

(25)

\begin{split}T_{s}(\Phi(u,v))=\alpha_{s}(u)+\sum_{l=1}^{d-k}v_{l}w_{l,s}(u),\end{split}

where

	$\displaystyle\alpha_{s}(u)=$	$\displaystyle\alpha(u)+e_{d+1}[\alpha(u)\cdot(N(s)-e_{d+1})-\alpha(s)\cdot N(s)],$
	$\displaystyle w_{l,s}(u)=$	$\displaystyle w_{l}(u)+e_{d+1}[w_{l}(u)\cdot(N(s)-e_{d+1})].$

Remark.

The new hypersurface $M_{s}:=T_{s}(M)$ is also $d-k$ ruled with a coordinate chart $\Phi_{s}:=T_{s}\circ\Phi$ if $s$ is sufficiently small. Since for all $l$ and $j$ , $p=T_{s}(\Phi(u,v))$ ,

\frac{\partial w_{l,s}}{\partial u_{j}}(u)\in T_{p}M_{s},

for $T_{s}$ defined in (25), the normal vectors of $T_{s}(M)$ are stable along the rulings, so $M_{s}$ is also a smooth hypersurface of constant rank $k$ .

4. Appendix

4.1. Hausdorff and Fourier dimensions

One notion of size often used in fractal geometry is the Hausdorff dimension, which is defined as follows. For a set $A\subset\mathbb{R}^{n}$ , $s,\delta>0$ ,

\mathcal{H}^{s}_{\delta}(A):=\inf\left\{\left.\sum_{j}\text{diam}(E_{j})^{s}\right|A\subset\bigcup_{j}E_{j},\text{diam}(E_{j})<\delta\right\},

and the $s$ -dimensional Hausdorff measure $\mathcal{H}^{s}(A):=\lim_{\delta\to 0}\mathcal{H}^{s}_{\delta}(A)$ . The Hausdorff dimension of $A$ is $\dim_{H}(A):=\sup\{s:\mathcal{H}^{s}(A)=\infty\}=\inf\{s:\mathcal{H}^{s}(A)=0\}.$

Frostman’s lemma [15, Theorem 2.7] offers an alternative characterization of the Hausdorff dimension. For a Borel set $A\subset\mathbb{R}^{n}$ ,

\dim_{H}(A)=\sup\{s\in[0,n]:\exists\mu\in\mathcal{M}(A),I_{s}(\mu):=\int|\widehat{\mu}(\xi)|^{2}|\xi|^{s-n}d\xi<\infty\},

where $I_{s}$ is the $s$ -energy of $\mu$ .

For $\mu\in\mathcal{M}(A)$ , if $\sup_{\xi\in\mathbb{R}^{n}}|\xi|^{\frac{s}{2}}|\widehat{\mu}(\xi)|<\infty$ for an $s>0$ , $I_{s_{0}}(\mu)<\infty$ when $0<s_{0}<s$ . From the characterization above, $\dim_{H}(A)\geq s$ . Consequently, $\dim_{F}(A)\leq\dim_{H}(A)$ , and the inequality is strict for some sets $A$ . The set $A$ is Salem if $\dim_{F}(A)=\dim_{H}(A)$ .

4.2. Fourier dimension of the Cartesian product

We present a lemma that yields the Fourier dimension of cylindrical hypersurfaces.

Lemma 4.1.

Suppose that $S\subset\mathbb{R}^{n}$ , $T\subset\mathbb{R}^{m}$ are compact, $\dim_{F}(S)<n$ , and $\dim_{F}(T)<m$ . Then,

a.

(26) $\dim_{F}(S\times T)=\min\{\dim_{F}(S),\dim_{F}(T)\},$
b.

(27) $\dim_{F}(S\times\mathbb{R}^{m})=\dim_{F}(S).$

We refer readers to the discussion in [4] on the Hausdorff dimension of the Cartesian product.

Proof.

For part a, let $s:=\dim_{F}(S)<n$ , $t:=\dim_{F}(T)<m$ . We write $x=(x_{1},x_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{m}$ .

•

First, we show that

\dim_{F}(S\times T)\geq\min\{s,t\}.

Let $\varepsilon>0$ . Let $\mu_{S}\in\mathcal{M}(S)$ , with $|\widehat{\mu_{S}}(\xi_{1})|\leq C_{1}|\xi_{1}|^{-\frac{s}{2}+\varepsilon}$ for $C_{1}>0$ , $\xi_{1}\in\mathbb{R}^{n}$ , $|\xi_{1}|\geq 1$ .

Similarly, let $\mu_{T}\in\mathcal{M}(T)$ , with $|\widehat{\mu_{T}}(\xi_{2})|\leq C_{2}|\xi_{2}|^{-\frac{t}{2}+\varepsilon}$ for $C_{2}>0$ , $\xi_{2}\in\mathbb{R}^{m}$ , $|\xi_{2}|\geq 1$ . We consider the measure $\mu_{S}\times\mu_{T}\in\mathcal{M}(S\times T)$ . For $\xi=(\xi_{1},\xi_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{m}$ ,

\begin{split}\widehat{\mu_{S}\times\mu_{T}}(\xi)&=\int_{\mathbb{R}^{n+m}}e^{-2\pi ix\cdot\xi}d(\mu_{S}\times\mu_{T})(x)\\ =&\left(\int_{\mathbb{R}^{n}}e^{-2\pi ix_{1}\cdot\xi_{1}}d\mu_{S}(x_{1})\right)\left(\int_{\mathbb{R}^{m}}e^{-2\pi ix_{2}\cdot\xi_{2}}d\mu_{T}(x_{2})\right)\\ =&\widehat{\mu_{S}}(\xi_{1})\widehat{\mu_{T}}(\xi_{2}).\end{split}

There are two cases to consider.

–

Case 1: $|\xi_{1}|\geq\frac{1}{2}|\xi|$ . Then,

|\widehat{\mu_{S}\times\mu_{T}}(\xi)|=|\widehat{\mu_{S}}(\xi_{1})\widehat{\mu_{T}}(\xi_{2})|\leq C_{1}|\xi_{1}|^{-\frac{s}{2}+\varepsilon}\leq C_{1}2^{\frac{s}{2}-\varepsilon}|\xi|^{-\frac{s}{2}+\varepsilon}.

–

Case 2: $|\xi_{2}|\geq\frac{1}{2}|\xi|$ . Similarly, $|\widehat{\mu_{S}\times\mu_{T}}(\xi)|\leq C_{2}2^{\frac{t}{2}-\varepsilon}|\xi|^{-\frac{t}{2}+\varepsilon}.$

This shows that $\dim_{F}(S\times T)\geq\min\{s-2\varepsilon,t-2\varepsilon\}$ , then we can let $\varepsilon\to 0$ .

•

Next, we show that

\dim_{F}(S\times T)\leq\min\{s,t\}.

Let $\mu\in\mathcal{M}(S\times T)$ , and suppose that $|\widehat{\mu}(\xi)|\leq C|\xi|^{-\frac{r}{2}}$ for $\xi=(\xi_{1},\xi_{2})\in\mathbb{R}^{n}\times\mathbb{R}^{m}$ . We will show that $r\leq s$ and $r\leq t$ .

–

Let $\pi_{S}:S\times T\to S$ be the projection map to the first $n$ coordinates and $\pi_{T}:S\times T\to T$ be the projection map to the last $m$ coordinates. Since $S$ and $T$ are compact, for $\mu\in\mathcal{M}(S\times T)$ ,

$\text{spt}\pi_{S}^{\#}\mu=\pi_{S}(\text{spt}\mu)\subset S,$

$\text{spt}\pi_{T}^{\#}\mu=\pi_{T}(\text{spt}\mu)\subset T.$

–

First, consider $\xi=(\xi_{1},0)$ (so $\xi_{2}=0$ ). Then,

(28)

\begin{split}\widehat{\mu}(\xi)&=\int e^{-2\pi ix\cdot\xi}d\mu(x)\\ &=\int e^{-2\pi ix_{1}\cdot\xi_{1}}d\mu(x)\\ &=\int e^{-2\pi i\pi_{S}(x)\cdot\xi_{1}}d\mu(x)\\ &=\int e^{-2\pi iy\cdot\xi_{1}}d\pi_{S}^{\#}\mu(y)\\ &=\widehat{\pi_{S}^{\#}\mu}(\xi_{1})\end{split}

Since $\pi_{S}^{\#}\mu\in\mathcal{M}(S)$ , $\dim_{F}(S)=s$ , $|\widehat{\pi_{S}^{\#}\mu}(\xi_{1})|=|\widehat{\mu}(\xi)|\leq C|\xi|^{-\frac{r}{2}}=C|\xi_{1}|^{-\frac{r}{2}}$ , then $r\leq s$ .

–

Next, consider $\xi=(0,\xi_{2})$ (so $\xi_{1}=0$ ). Similar to (28), $|\widehat{\mu}(\xi)|=|\widehat{\pi_{T}^{\#}\mu}(\xi_{2})|$ . As $\pi_{T}^{\#}\mu\in\mathcal{M}(T)$ , $\dim_{F}(T)=t$ , $|\widehat{\pi_{T}^{\#}\mu}(\xi_{2})|=|\widehat{\mu}(\xi)|\leq C|\xi|^{-\frac{r}{2}}=C|\xi_{2}|^{-\frac{r}{2}}$ , we have $r\leq t$ .

The proof for part b) is similar, where we can take $\mu_{T}\in\mathcal{S}(\mathbb{R}^{m})$ . ∎

4.3. Reduction to compactly supported measure

We present a lemma that allows us to assume that the measure $\mu\in\mathcal{M}(M)$ we study has smaller support in a smaller closed set on the manifold $M$ .

Lemma 4.2.

[3, Theorem 1] Suppose that $\mu_{0}\in\mathcal{M}(\mathbb{R}^{n})$ , $\sup_{\xi\in\mathbb{R}^{n}}|\xi|^{\alpha}|\widehat{\mu_{0}}(\xi)|<\infty$ for $\alpha>0$ . Let $f\in\mathcal{S}(\mathbb{R}^{n})$ with $f\geq 0$ , and $\mu\in\mathcal{M}(\mathbb{R}^{n})$ such that $d\mu=fd\mu_{0}$ . Then, $|\widehat{\mu}(\xi)|\leq c_{\mu}|\xi|^{-\alpha}$ for a $c_{\mu}>0$ .

Proof.

Note that

\begin{split}|\widehat{\mu}(\xi)|&=|\widehat{\mu_{0}}*\widehat{f}(\xi)|\\ &=\left|\int\widehat{\mu_{0}}(\xi-\eta)\widehat{f}(\eta)d\eta\right|\\ &\leq\left|\int_{|\eta|\leq\frac{|\xi|}{2}}\widehat{\mu_{0}}(\xi-\eta)\widehat{f}(\eta)d\eta\right|+\left|\int_{|\eta|\geq\frac{|\xi|}{2}}\widehat{\mu_{0}}(\xi-\eta)\widehat{f}(\eta)d\eta\right|.\end{split}

For the first integral, since $|\eta|\leq\frac{|\xi|}{2}$ , $|\xi-\eta|\geq\frac{|\xi|}{2}$ , and $\widehat{f}$ is integrable, the integral is bounded above by a constant multiple of $|\xi|^{-\alpha}$ . For the second integral, we apply the bounds $\left\lVert\widehat{\mu_{0}}\right\rVert_{L^{\infty}}=1$ and $\int_{|\eta|\geq\frac{|\xi|}{2}}|\widehat{f}(\eta)|d\eta\leq c_{m}|\xi|^{-m}$ for $m\in\mathbb{N}$ with a $c_{m}>0$ since $\widehat{f}\in\mathcal{S}(\mathbb{R}^{n})$ . If we choose $m>\alpha$ , the sum of two bounds is bounded by a constant multiple of $|\xi|^{-\alpha}$ for large $|\xi|$ . ∎

4.4. Morse’s lemma

The version stated below generalizes the one shown in [18, Section 8.2].

Lemma 4.3.

Suppose that $f\in C^{\infty}(\mathbb{R}^{d}\times\mathbb{R}^{d})$ , $f(0,t)=0$ , $\nabla_{x}f(0,t)=0$ , and that $\Delta_{x}f(x,t)$ has a constant rank of $n$ . Then, there exist neighborhoods $V,W$ of $0$ and a smooth $\tau:V\times W\to\mathbb{R}^{d}$ such that $\tau(0,t)=0$ , $\det{\bf J}_{x}\tau(x,t)\neq 0$ , and

(29)

f(x,t)=Q_{m,n}(\tau(x,t)),

where

(30)

Q_{m,n}(y):=\sum_{j=1}^{m}y_{j}^{2}-\sum_{j=m+1}^{n}y_{j}^{2}

with $0\leq m\leq n\leq d$ .

Remark.

The number of positive eigenvalues of the matrix $\Delta_{x}f(0,0)$ is $m$ . By differentiating (29) twice,

(31)

[{\bf J}_{x}\tau(0,t)]^{T}\Delta Q_{m,n}(0){\bf J}_{x}\tau(0,t)=\Delta_{x}f(0,t).

Proof.

We claim that the function $\tau$ can be expressed as

(32)

\tau=L\circ\tau_{d}\circ\cdots\circ\tau_{1},

where each $\tau_{r}$ is a change of variables in the first $d$ coordinates and where $L$ is a permutation of the same coordinates. $\tau_{r}$ is constructed inductively as follows: suppose that at step $r$ , we have

\widetilde{g_{r}}(x,t):=f(\tau_{1}^{-1}\circ\cdots\circ\tau_{r-1}^{-1}(x,t))=\pm[x_{1}]^{2}\pm\cdots\pm[x_{r-1}]^{2}+\sum_{j,k\geq r}^{d}x_{j}x_{k}\widetilde{f_{j,k}}(x,t),

where

\widetilde{f_{j,k}}(x,t)=\int_{0}^{1}(1-s)\frac{\partial^{2}\widetilde{g_{r}}}{\partial x_{j}\partial x_{k}}(sx,t)ds,\widetilde{f_{j,k}}(0,t)=\frac{1}{2}\frac{\partial^{2}\widetilde{g_{r}}}{\partial x_{j}\partial x_{k}}(0,t).

When $r>n$ , since $\Delta_{x}\widetilde{g_{r}}(x,t)$ only has rank $n$ , all $\widetilde{f_{j,k}}(x,t)=0$ , and we are done. Otherwise, there exists an orthonormal matrix $O_{r}$ , which is a linear change in the variables $x_{r},\cdots,x_{d}$ , such that

g_{r}(y,t):=f(\tau_{1}^{-1}\circ\cdots\circ\tau_{r-1}^{-1}(O_{r}y,t))=\pm[y_{1}]^{2}\pm\cdots\pm[y_{r-1}]^{2}+\sum_{j,k\geq r}^{d}y_{j}y_{k}f_{j,k}(y,t),

where

f_{j,k}(y,t)=\int_{0}^{1}(1-s)\frac{\partial^{2}g_{r}}{\partial y_{j}\partial y_{k}}(sy,t)ds,f_{j,k}(0,t)=\frac{1}{2}\frac{\partial^{2}g_{r}}{\partial y_{j}\partial y_{k}}(0,t)

and the additional condition that $f_{r,r}(0,t)\neq 0$ . Then, for each $t$ , we can perform a change of variables from $y$ to $z$ such that $z_{j}=y_{j}$ for $j\neq r$ , and

(33)

z_{r}=[\pm f_{r,r}(y,t)]^{\frac{1}{2}}\left[y_{r}+\sum_{j>r}\frac{y_{j}f_{j,r}(y,t)}{\pm f_{r,r}(y,t)}\right],

where $\pm$ is the sign of $f_{r,r}(0,t)$ . This change of variables can be expressed as $z=\sigma_{r}(y,t)$ for $\sigma_{r}:V_{r}\times W_{r}\to\mathbb{R}^{d}$ , where $V_{r},W_{r}$ are neighborhoods of $0$ , $f_{r,r}$ does not change sign on $V_{r}\times W_{r}$ , and

\det{\bf J}_{y}\sigma_{r}(y,t)\neq 0.

Then, we let

(34)

\tau_{r}(x,t)=(\sigma_{r}(O_{r}^{-1}x,t),t),

and proceed to the next step by noting that $\tau_{r}^{-1}$ exists in a neighborhood of $0$ .

From the construction above, there exist $V,W$ neighborhoods of $0$ , such that for $(x,t)=(x,t)\in V\times W$ , for all $r$ from $1$ to $d$ , $f_{r,r}(y,t)$ does not change sign, and

\det{\bf J}_{y}\sigma_{r}(y,t)\neq 0.

Therefore, $\tau$ is defined on $V\times W$ . ∎

Let $\tau_{t}(x)=\tau(x,t)$ . For $k\geq 0$ , $t\in W$ , it is possible to bound $\left\lVert{\bf J}\tau_{t}\right\rVert_{C^{k}(V)}$ and $\inf\{|\det{\bf J}\tau_{t}(x)|x\in V\}$ via $\left\lVert f_{t}\right\rVert_{C^{k+2}}$ , $|\det{\bf J}\tau_{t}(0)|$ , and the size of $V$ via (32), (33), and (34).

4.5. Oscillatory Integrals

We refer readers to the complete proof of oscillatory integrals results in [18]. In this section, we outline the key steps and bounds of the error terms.

For $f\in C^{\infty}(\mathbb{R}^{n})$ , $U$ open in $\mathbb{R}^{n}$ , and $k\geq 0$ , we denote $\left\lVert f\right\rVert_{C^{k}(U)}$ , or $\left\lVert f\right\rVert_{C^{k}}$ if the implication of the open set $U$ is clear, as the quantity

\sum_{|\beta|\leq k}\left\lVert\frac{\partial^{|\beta|}}{\partial y^{\beta}}f\right\rVert_{L^{\infty}(U)}.

Proposition 4.4.

[18, Chapter 8, Propositions 1 and 4] (Localization)

(Single variable edition) For $N\in\mathbb{N}$ , there exist $C_{N}>0$ , $p_{N},q_{N}\in\mathbb{N}$ such that, for $\phi\in C^{\infty}(\mathbb{R})$ , $\psi\in C^{\infty}_{0}(\mathbb{R})$ with $|\phi^{\prime}(x)|\geq 1$ in $\text{spt }\psi$ , $\lambda\geq 1$ ,

\left|\int e^{i\lambda\phi(x)}\psi(x)dx\right|\leq C_{N}\lambda^{-N}|\text{spt }\psi|\left\lVert\psi\right\rVert_{C^{N}}^{p_{N}}\left\lVert\phi^{\prime\prime}\right\rVert_{C^{N-1}}^{q_{N}}

(Multivariate edition) For $N\in\mathbb{N}$ , there exist $q_{N},r_{N}\in\mathbb{N}$ , such that for $\phi\in C^{\infty}(\mathbb{R}^{n})$ , $\psi\in C^{\infty}_{0}(\mathbb{R}^{n})$ with $|\nabla\phi(x)|\geq 1$ in $\text{spt }\psi$ , $\Lambda$ being the maximum absolute values of the eigenvalues of $\Delta\phi(x^{\prime})$ for all $x^{\prime}\in\text{spt }\psi$ , $\lambda\geq 1$ , there exists $C_{N,\psi}>0$ with

\left|\int e^{i\lambda\phi(x)}\psi(x)dx\right|\leq C_{N,\psi}\lambda^{-N}(1+\Lambda)^{r_{N}}\max_{\xi\in\mathbb{S}^{n-1}}\left\lVert\frac{\partial^{2}\phi}{\partial\xi^{2}}\right\rVert_{C^{N-1}}^{q_{N}}.

Proof.

Using integration by parts $N$ times, the integral can be written as

\int e^{i\lambda\phi(x)}(^{t}D)^{N}\{\psi(x)\}dx

with ${}^{t}Dg=i\lambda^{-1}\frac{d}{dx}\left(\frac{g}{\phi^{\prime}}\right)$ . Note that

\lambda^{N}(^{t}D)^{N}g=\frac{g^{(N)}}{(\phi^{\prime})^{N}}+\sum_{r=1}^{N}(\phi^{\prime})^{-(N+r)}F_{N,r}(g,g^{\prime},\cdots,g^{(N-1)},\phi^{\prime\prime},\cdots,\phi^{(N+1)})

for some polynomials $F_{N,r}$ . Therefore, the integral is bounded above by

C_{N}\lambda^{-N}|\text{spt }\psi|\left\lVert\psi\right\rVert_{C^{N}}^{p_{N}}\left\lVert\phi^{\prime\prime}\right\rVert_{C^{N-1}}^{q_{N}}

for $C_{N}>0$ , $p_{N},q_{N}\in\mathbb{N}$ .

For each $x_{0}$ in the support of $\psi$ , there is a unit vector $\xi_{0}=(\nabla\phi)(x_{0})/\left\lVert(\nabla\phi)(x_{0})\right\rVert$ and a ball centered $B(x_{0})$ at $x_{0}$ with

(35)

\xi_{0}\cdot(\nabla\phi)(x)\geq 2^{-1}

if $x\in B(x_{0})$ . We claim that $B(x_{0})$ can be chosen with the radius bounded below by a uniform constant depending on $\phi$ . By the mean-value theorem, there exists $x^{\prime}$ on the line segment joining $x_{0}$ and $x$ , such that

|\xi_{0}\cdot(\nabla\phi)(x)-\xi_{0}\cdot(\nabla\phi)(x_{0})|=|(x-x_{0})\Delta\phi(x^{\prime})\xi_{0}|.

|x-x_{0}|\leq(2(1+\Lambda))^{-1},

then (35) holds.

We choose a finite cover $\{B_{k}\}\subset\{B(x_{0})\}$ , and write $\psi=\sum_{k}\psi_{k}$ for $\text{spt }\psi_{k}\subset B_{k}$ . For each $B_{k}$ , we choose a coordinate system where $x_{1}$ is parallel to $\xi_{k}$ . Then,

\int e^{i\lambda\phi(x)}\psi_{k}(x)dx=\int\left(\int e^{i\lambda\phi(x_{1},\cdots,x_{n})}\psi_{k}(x_{1},\cdots,x_{n})dx_{1}\right)dx_{2}\cdots dx_{n},

and we can apply part a) to the inner integral and bound it above by

C_{N}\lambda^{-N}|\text{spt }\psi_{k}|\left\lVert\psi_{k}\right\rVert_{C^{N}}^{p_{N}}\left\lVert\frac{\partial^{2}\phi}{\partial x_{1}^{2}}\right\rVert_{C^{N-1}}^{q_{N}}.

When all $k$ ’s are summed, a bound on the original integral is

C_{N}^{\prime}\lambda^{-N}|\text{spt }\psi|\left\lVert\psi\right\rVert_{C^{N}}^{p_{N}}(1+\Lambda)^{r_{N}}\max_{\xi\in\mathbb{S}^{n-1}}\left\lVert\frac{\partial^{2}\phi}{\partial\xi^{2}}\right\rVert_{C^{N-1}}^{q_{N}}

for $C_{N}^{\prime}>0$ , and $r_{N}\in\mathbb{N}$ . ∎

Lemma 4.5.

[18, Chapter 8, Proposition 6] Let $Q_{m,n}(y)$ be defined in (30).

a.

If $\eta\in C^{\infty}_{0}(\mathbb{R}^{n})$ ,

(36) $\left|\int_{\mathbb{R}^{n}}e^{i\lambda Q_{m,n}(y)}y^{l}\eta(y)dy\right|\leq A_{l}\lambda^{-\frac{n+|l|}{2}},$

where $A_{l}>0$ , $l\in\mathbb{Z}^{n}$ , $l_{j}\geq 0$ , and $|l|=\sum_{j=1}^{n}l_{j}$ .
b.

If $g\in\mathcal{S}(\mathbb{R}^{n})$ and there exists $\delta>0$ , such that $g(y)=0$ for $y\in B(0,\delta)$ , then for $N\in\mathbb{N}$ , there exists $B_{N}>0$ such that

(37) $\left|\int_{\mathbb{R}^{n}}e^{i\lambda Q_{m,n}(y)}g(y)dy\right|\leq B_{N}\lambda^{-N}.$

Proof.

Consider the cones

\Gamma_{k}=\left\{y\in\mathbb{R}^{n}||y_{k}|^{2}\geq\frac{|y|^{2}}{2n}\right\},

and

\Gamma_{k}^{0}=\left\{y\in\mathbb{R}^{n}||y_{k}|^{2}\geq\frac{|y|^{2}}{n}\right\}.

Since $\cup_{k=1}^{n}\Gamma_{k}^{0}=\mathbb{R}^{n}$ , there are functions $\{\Omega_{k}\}_{1\leq k\leq n}$ such that each $\Omega_{k}$ is homogeneous of degree $0$ , smooth away from the origin, $0\leq\Omega_{k}\leq 1$ with

\sum_{k=1}^{n}\Omega_{k}(x)=1

for $x\neq 0$ , and each $\Omega_{k}$ is supported in $\Gamma_{k}$ . Then,

\int_{\mathbb{R}^{n}}e^{i\lambda Q_{m,n}(y)}y^{l}\eta(y)dy=\sum_{k=1}^{n}\int_{\Gamma_{k}}e^{i\lambda Q_{m,n}(y)}y^{l}\eta(y)\Omega_{k}(y)dy.

In the cone $\Gamma_{k}$ , we will show that there exists $A_{l,k}>0$ such that

(38)

\left|\int_{\Gamma_{k}}e^{i\lambda Q_{m,n}(y)}y^{l}\eta(y)\Omega_{k}(y)dy\right|\leq A_{l,k}\lambda^{-\frac{n+|l|}{2}}.

Summing over all $k$ yields (36). Let $\alpha\in C^{\infty}(\mathbb{R}^{n})$ , such that $\alpha(y)=1$ for $|y|\leq 1$ , and $\alpha(y)=0$ for $|y|\geq 2$ . Then, for $\epsilon>0$ ,

\begin{split}&\int_{\Gamma_{k}}e^{i\lambda Q_{m,n}(y)}y^{l}\eta(y)\Omega_{k}(y)dy\\ =&\int_{\Gamma_{k}}e^{i\lambda Q_{m,n}(y)}y^{l}\eta(y)\Omega_{k}(y)\alpha(\epsilon^{-1}y)dy+\int_{\Gamma_{k}}e^{i\lambda Q_{m,n}(y)}y^{l}\eta(y)\Omega_{k}(y)[1-\alpha(\epsilon^{-1}y)]dy.\end{split}

For the first integral,

\left|\int_{\Gamma_{k}}e^{i\lambda Q_{m,n}(y)}y^{l}\eta(y)\Omega_{k}(y)\alpha(\epsilon^{-1}y)dy\right|\lesssim\left\lVert\eta\right\rVert_{L^{\infty}}\epsilon^{|l|+1}.

Let $N\in\mathbb{N}$ . Using integration by parts $N$ times, the second integral can be written as

(39)

\int_{\Gamma_{k}}e^{i\lambda Q_{m,n}(y)}(^{t}D_{k})^{N}\{y^{l}\eta(y)\Omega_{k}(y)[1-\alpha(\epsilon^{-1}y)]\}dy

with ${}^{t}D_{k}g=s_{k}(2i\lambda)^{-1}\frac{\partial}{\partial y_{k}}\left(\frac{g}{y_{k}}\right)$ for a differentiable function $g$ , and $s_{k}=-1$ if $k\leq m$ , $s_{k}=1$ if $k\geq m+1$ . Note that

(40)

(^{t}D_{k})^{N}g=\lambda^{-N}\sum_{r=0}^{N}a^{(m,k)}_{N,r}y_{k}^{r-2N}\frac{\partial^{r}g}{\partial y_{k}^{r}}

for $a^{(m,k)}_{N,r}\in\mathbb{C}$ . When we apply (40) and the product rule of the derivative to expand (39), we obtain a summation of terms where a term, ignoring the constant, is

\lambda^{-N}\int_{\Gamma_{k}\cap B(0,\epsilon)^{c}}e^{i\lambda Q_{m,n}(y)}y_{k}^{r-2N}\left[\frac{\partial^{r_{1}}}{\partial y_{k}^{r_{1}}}y^{l}\right]\left[\frac{\partial^{r_{2}}}{\partial y_{k}^{r_{2}}}\eta(y)\right]\left[\frac{\partial^{r_{3}}}{\partial y_{k}^{r_{3}}}\Omega_{k}(y)\right]\left[\frac{\partial^{r_{4}}}{\partial y_{k}^{r_{4}}}[1-\alpha(\epsilon^{-1}y)]\right]dy

for $r_{1},r_{2},r_{3},r_{4}\geq 0$ , $r_{1}+r_{2}+r_{3}+r_{4}=r\leq N$ . We note that $\frac{\partial^{r_{3}}\Omega_{k}}{\partial y_{k}^{r_{3}}}$ is a homogeneous function of degree $-r_{3}$ . When $|l|-N<-n$ , the term above is bounded by a constant multiple of

\begin{split}&\lambda^{-N}\epsilon^{-r_{4}}\int_{\Gamma_{k}\cap B(0,\epsilon)^{c}}|y|^{r-2N+|l|-r_{1}-r_{3}}\left\lVert\frac{\partial^{r_{2}}\eta}{\partial y_{k}^{r_{2}}}\right\rVert_{L^{\infty}}\left\lVert\frac{\partial^{r_{4}}\alpha}{\partial y_{k}^{r_{4}}}\right\rVert_{L^{\infty}}dy\\ \lesssim&\lambda^{-N}\epsilon^{-r_{4}}\epsilon^{r-2N+|l|-r_{1}-r_{3}+n}\left\lVert\eta\right\rVert_{C^{r_{2}}}\left\lVert\alpha\right\rVert_{C^{r_{4}}}\\ \lesssim&\lambda^{-N}\epsilon^{|l|-2N+r-r_{1}-r_{3}-r_{4}+n}\left\lVert\eta\right\rVert_{C^{r_{2}}}\left\lVert\alpha\right\rVert_{C^{r_{4}}}.\end{split}

Then we obtain (38) by setting $\epsilon=\lambda^{-\frac{1}{2}}$ . $A_{k}$ depends on $\left\lVert\eta\right\rVert_{C^{N}}$ .

The proof for b) is similar to that for a). We use the same cone $\Gamma_{k}$ and functions $\Omega_{k}$ , $\alpha$ . It suffices to show

(41)

\left|\int_{\Gamma_{k}}e^{i\lambda Q_{m,n}(y)}g(y)\Omega_{k}(y)dy\right|\leq B_{N,k}\lambda^{-N}\

for a $B_{N,k}>0$ . If $2\epsilon<\delta$ ,

\int_{\Gamma_{k}}e^{i\lambda Q_{m,n}(y)}g(y)\Omega_{k}(y)dy=\int_{\Gamma_{k}\cap B(0,\epsilon)^{c}}e^{i\lambda Q_{m,n}(y)}g(y)\Omega_{k}(y)[1-\alpha(\epsilon^{-1}y)]dy.

Then, we apply integration by parts $N$ times to obtain

(42)

\int_{\Gamma_{k}\cap B(0,\epsilon)^{c}}e^{i\lambda Q_{m,n}(y)}(^{t}D)^{N}\left\{g(y)\Omega_{k}(y)[1-\alpha(\epsilon^{-1}y)]\right\}dy.

After we apply (40) and the product rule of the derivative to expand (42), we obtain a summation of terms where a term, ignoring the constant, is

\lambda^{-N}\int_{\Gamma_{k}}e^{i\lambda Q_{m,n}(y)}y_{k}^{r-2N}\left[\frac{\partial^{r_{1}}}{\partial y_{k}^{r_{1}}}g(y)\right]\left[\frac{\partial^{r_{2}}}{\partial y_{k}^{r_{2}}}\Omega_{k}(y)\right]\left[\frac{\partial^{r_{3}}}{\partial y_{k}^{r_{3}}}[1-\alpha(\epsilon^{-1}y)]\right]dy

for $r_{1},r_{2},r_{3}\geq 0$ , $r_{1}+r_{2}+r_{3}=r\leq N$ . We note that $\frac{\partial^{r_{2}}\Omega_{k}}{\partial y_{k}^{r_{2}}}$ is a homogeneous function of degree $-r_{2}$ . When $-N<-n$ , the term above is bounded by a constant multiple of

\begin{split}&\lambda^{-N}\epsilon^{-r_{3}}\int_{\Gamma_{k}\cap B(0,\epsilon)^{c}}|y|^{r-2N-r_{2}}\left\lVert\frac{\partial^{r_{1}}g}{\partial y_{k}^{r_{1}}}\right\rVert_{L^{\infty}}\left\lVert\frac{\partial^{r_{3}}\alpha}{\partial y_{k}^{r_{3}}}\right\rVert_{L^{\infty}}dy\\ \lesssim&\lambda^{-N}\epsilon^{-r_{3}}\epsilon^{r-2N-r_{2}+n}\left\lVert g\right\rVert_{C^{r_{1}}}\left\lVert\alpha\right\rVert_{C^{r_{3}}}\\ \lesssim&\lambda^{-N}\epsilon^{r-2N-r_{2}-r_{3}+n}\left\lVert g\right\rVert_{C^{r_{1}}}\left\lVert\alpha\right\rVert_{C^{r_{3}}}.\end{split}

Then, we obtain (41) by setting $\epsilon=\frac{\delta}{3}$ . ∎

Theorem 4.6.

[18, Chapter 8, Proposition 6]

Let $Q_{m,n}(y)$ be defined as in (30). Let

I_{m}(\lambda;\psi):=\int_{\mathbb{R}^{n}}e^{i\lambda Q_{m,n}(y)}\psi(y)dy,

where $\psi$ is supported in a small neighborhood of $0$ , and $\lambda_{0}>1$ . For $\lambda\geq\lambda_{0}$ ,

(43)

\left|I_{m}(\lambda;\psi)-(-1)^{\frac{n-m}{2}}(\pi i)^{\frac{n}{2}}\psi(0)\lambda^{-\frac{n}{2}}\right|\leq D\lambda^{-\frac{n+1}{2}},

where $D$ depends on $\lambda_{0}$ , the size of $\text{spt }\psi$ , and $\left\lVert\psi\right\rVert_{C^{n+3}}$ .

Let $\phi,\psi\in C^{\infty}(\mathbb{R}^{n})$ . Suppose that $\phi$ has only one non-degnerate critical point $z_{0}$ in $\text{spt }\psi$ and $\phi(z_{0})=0$ . Let

I(\lambda;\phi,\psi):=\int_{\mathbb{R}^{n}}e^{i\lambda\phi(z)}\psi(z)dz,

$\lambda_{0}>1$ , and $c=\det\Delta\phi(z_{0})$ . For $\lambda\geq\lambda_{0}$ , there exist $0\leq m\leq n$ and $D=D(\lambda_{0},\phi,\psi)>0$ , such that

\left|I(\lambda;\phi,\psi)-(-1)^{\frac{n-m}{2}}(2\pi i)^{\frac{n}{2}}c^{-\frac{1}{2}}\psi(z_{0})\lambda^{-\frac{n}{2}}\right|\leq D(\lambda_{0},\phi,\psi)|\lambda|^{-\frac{n+1}{2}}.

The error $D(\lambda_{0},\phi,\psi)$ depends on $\lambda_{0}$ , $c$ , the size of $\text{spt }\psi$ , and $\left\lVert\phi\right\rVert_{C^{n+6}}$ , $\left\lVert\psi\right\rVert_{C^{n+3}}$ .

Proof.

Step 1: We have

\int_{-\infty}^{\infty}e^{i\lambda x^{2}}e^{-x^{2}}dx=(1-i\lambda)^{-\frac{1}{2}}\int_{-\infty}^{\infty}e^{-x^{2}}dx,

and $\int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}$ . We can fix the principal branch of $z^{-\frac{1}{2}}$ in the complex plane slit along the negative real-axis. Therefore,

(44)

\begin{split}\int_{\mathbb{R}^{n}}e^{i\lambda Q_{m,n}(y)}e^{-|y|^{2}}dy=&\left(\prod_{j=1}^{m}\int_{\mathbb{R}}e^{i\lambda y_{j}^{2}}e^{-y_{j}^{2}}dy_{j}\right)\left(\prod_{j=m+1}^{n}\int_{\mathbb{R}}e^{-i\lambda y_{j}^{2}}e^{-y_{j}^{2}}dy_{j}\right)\\ =&\pi^{\frac{n}{2}}(1-i\lambda)^{-\frac{m}{2}}(1+i\lambda)^{-\frac{n-m}{2}}\\ =&\pi^{\frac{n}{2}}\lambda^{-\frac{n}{2}}(\lambda^{-1}-i)^{-\frac{m}{2}}(\lambda^{-1}+i)^{-\frac{n-m}{2}}.\end{split}

We write the power series expansion of $f_{m}(w)=(w-i)^{-\frac{m}{2}}(w+i)^{-\frac{n-m}{2}}$ at $0$ as $\sum_{k=0}^{\infty}a_{k}w^{k},$ where $a_{0}=(-1)^{\frac{n-m}{2}}i^{\frac{n}{2}}$ . Let $\gamma$ be a line segment from $0$ to $\lambda^{-1}$ and $b_{m}=\sup_{|w|=\lambda_{0}^{-1}}|f_{m}^{\prime}(w)|$ . Then for $\lambda\geq\lambda_{0}>1$ , the error of approximation by the constant term is bounded by

(45)

\begin{split}|f_{m}(\lambda^{-1})-a_{0}|\leq&\left|\int_{\gamma}f_{m}^{\prime}(w)dw\right|\\ \leq&|\gamma|\sup_{|w|=\lambda^{-1}}|f_{m}^{\prime}(w)|\\ \leq&b_{m}\lambda^{-1}.\end{split}

The integral form of the Taylor remainder and the maximum modulus principle are used. Putting everything together, for $\lambda\geq\lambda_{0}$ ,

(46)		$\displaystyle\left\|\int_{\mathbb{R}^{n}}e^{i\lambda Q_{m,n}(y)}e^{-\|y\|^{2}}dy-a_{0}\pi^{\frac{n}{2}}\lambda^{-\frac{n}{2}}\right\|$
	$\displaystyle=$	$\displaystyle\left\|\pi^{\frac{n}{2}}\lambda^{-\frac{n}{2}}(\lambda^{-1}-i)^{-\frac{m}{2}}(\lambda^{-1}+i)^{-\frac{n-m}{2}}-a_{0}\pi^{\frac{n}{2}}\lambda^{-\frac{n}{2}}\right\|$	by (44)
	$\displaystyle=$	$\displaystyle\pi^{\frac{n}{2}}\lambda^{-\frac{n}{2}}\left\|(\lambda^{-1}-i)^{-\frac{m}{2}}(\lambda^{-1}+i)^{-\frac{n-m}{2}}-a_{0}\right\|$
	$\displaystyle\leq$	$\displaystyle b_{m}\pi^{\frac{n}{2}}\lambda^{-\frac{n+2}{2}}$	$\displaystyle\text{ by (\ref{eq_fm_a0_d})}.$

Step 2: To obtain (43), we write $e^{|y|^{2}}\psi(y)=\psi(0)+\sum_{y=1}^{n}y_{j}R_{j}(y)$ , where $R_{j}\in C_{0}^{\infty}(\mathbb{R}^{n})$ . Let $\bar{\psi}\in C^{\infty}_{0}(\mathbb{R}^{n})$ , with $\bar{\psi}(y)=1$ on the support of $\psi$ . To apply the results from the previous steps, we write

\begin{split}&\int_{\mathbb{R}^{n}}e^{i\lambda Q_{m,n}(y)}\psi(y)dy\\ =&\int e^{i\lambda Q_{m,n}(y)}e^{-|y|^{2}}[e^{|y|^{2}}\psi(y)]\bar{\psi}(y)dy\\ =&\int_{\mathbb{R}^{n}}e^{i\lambda Q_{m,n}(y)}e^{-|y|^{2}}\left[\psi(0)+\sum_{j=1}^{n}y_{j}R_{j}(y)\right]\bar{\psi}(y)dy\\ =&\int e^{i\lambda Q_{m,n}(y)}e^{-|y|^{2}}\psi(0)\bar{\psi}(y)dy+\sum_{j=1}^{n}\int e^{i\lambda Q_{m,n}(y)}y_{j}e^{-|y|^{2}}R_{j}(y)\bar{\psi}(y)dy\\ =&\psi(0)\int e^{i\lambda Q_{m,n}(y)}e^{-|y|^{2}}dy+\psi(0)\int e^{i\lambda Q_{m,n}(y)}e^{-|y|^{2}}[1-\bar{\psi}(y)]dy\\ &+\sum_{j=1}^{n}\int e^{i\lambda Q_{m,n}(y)}y_{j}e^{-|y|^{2}}R_{j}(y)\bar{\psi}(y)dy.\end{split}

Then we obtain (43) by applying (46) to the first integral, (37) to the second integral, and (36) to each term in the summation. We note that the error $D$ depends on $\lambda_{0}$ , the size of the support of $\psi$ , and $\left\lVert\psi\right\rVert_{C^{n+3}}$ .

Let $V$ be a neighborhood of $z_{0}$ obtained from applying Morse’s lemma (Lemma 4.3) to $\phi$ , and we write $\psi=\psi_{1}+\psi_{2}$ , where $\text{spt }\psi_{1}\subset V\cap\text{spt }\psi$ , and $\text{spt }\psi_{2}$ does not contain $z_{0}$ . Then, we write

\int_{\mathbb{R}^{n}}e^{i\lambda\phi(z)}\psi(z)dz=\int_{\mathbb{R}^{n}}e^{i\lambda\phi(z)}\psi_{1}(z)dz+\int_{\mathbb{R}^{n}}e^{i\lambda\phi(z)}\psi_{2}(z)dz.

•

By Morse’s lemma (Lemma 4.3), there exists a diffeomorphism $\tau:V\to U$ , where $V$ is a neighborhood of $z_{0}$ in the $z$ -space, and $U$ is a neighborhood of $0$ in the $y$ -space, such that $\tau(z_{0})=0$ and

\phi(z)=Q_{m,n}(\tau(z)).

By a change of variables $z=\tau^{-1}(y)$ ,

\begin{split}\int_{\mathbb{R}^{n}}e^{i\lambda\phi(z)}\psi_{1}(z)dz=&\int e^{i\lambda Q_{m,n}(y)}\psi_{1}(\tau^{-1}(y))|\det{\bf J}\tau^{-1}(y)|dy\\ =&I_{m}(\lambda;(\psi_{1}\circ\tau^{-1})\cdot|\det{\bf J}\tau^{-1}|).\end{split}

We can apply part a) to the integral above. $D$ depends on $\lambda_{0}$ , the size of support of $\psi_{1}\circ\tau^{-1}$ , and the $C^{n+3}$ norms of $(\psi_{1}\circ\tau^{-1})\cdot|\det{\bf J}\tau^{-1}|$ . We note that the $L^{\infty}$ norm of the $k^{\text{th}}$ partial derivative ( $0\leq k\leq n+3$ ) of $(\psi_{1}\circ\tau^{-1})\cdot|\det{\bf J}\tau^{-1}|$ can be bounded by $C^{n+3}$ norms of $\psi$ , $|\det{\bf J}\tau|$ , and $\inf\{|\det{\bf J}\tau(z)|z\in\text{spt}\psi\}$ . We note that $|\det{\bf J}\tau(0)|^{2}=\frac{c}{2^{n}}$ by (31), and we can bound the error in terms of $\phi$ with the fact that $\left\lVert\det{\bf J}\tau\right\rVert_{C^{n+3}(V)}$ and $\inf\{|\det{\bf J}\tau(z)|z\in\text{spt}\psi\}$ depend on $\left\lVert\phi\right\rVert_{C^{n+5}(V)}$ , $c$ , and the size of $\text{spt }\psi_{1}$ from the discussion of Morse’s lemma in Section 4.4.

•

For the second integral, we apply Proposition 4.4. ∎

References

[1] Christian Bluhm. Random recursive construction of Salem sets. Ark. Mat., 34(1):51–63, 1996.
[2] Manfredo P. d. Carmo. Differential geometry of curves and surfaces. Prentice-Hall, Englewood Cliffs, NJ, 1976.
[3] Fredrik Ekström, Tomas Persson, and Jörg Schmeling. On the Fourier dimension and a modification. J. Fractal Geom., 2(3):309–337, 2015.
[4] Kenneth Falconer. Fractal geometry : Mathematical foundations and applications. John Wiley and Sons, Incorporated, Hoboken, NJ, third edition, 2014.
[5] Jonathan M. Fraser, Terence L. J. Harris, and Nicholas G. Kroon. On the Fourier dimension of $(d,k)$ -sets and Kakeya sets with restricted directions. Math. Z., 301(3):2497–2508, 2022.
[6] Robert Fraser and Kyle Hambrook. Explicit Salem sets in $\mathbb{R}^{n}$ . Adv. Math., 416:Paper No. 108901, 23, 2023.
[7] Kanghui Guo. On the $p$ -thin problem for hypersurfaces of $\mathbb{R}^{n}$ with zero Gaussian curvature. Canad. Math. Bull., 36(1):64–73, 1993.
[8] Philip Hartman. On isometric immersions in Euclidean space of manifolds with non-negative sectional curvatures. Trans. Amer. Math. Soc., 115:94–109, 1965.
[9] Edmund Hlawka. über Integrale auf konvexen Körpern. I. Monatsh. Math., 54:1–36, 1950.
[10] Jean-Pierre Kahane. Some random series of functions, volume 5 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, UK, second edition, 1985.
[11] Robert Kaufman. On the theorem of Jarník and Besicovitch. Acta Arith., 39(3):265–267, 1981.
[12] Izabella Laba and Malabika Pramanik. Arithmetic progressions in sets of fractional dimension. Geometric and Functional Analysis, 19(2):429–456, 2009.
[13] Yiyu Liang and Malabika Pramanik. Fourier dimension and avoidance of linear patterns. Advances in Mathematics, 399:108252, 2022.
[14] Walter Littman. Fourier transforms of surface-carried measures and differentiability of surface averages. Bull. Amer. Math. Soc., 69:766–770, 1963.
[15] Pertti Mattila. Fourier analysis and Hausdorff dimension, volume 150. Cambridge University Press, Cambridge, UK, 1 edition, 2015.
[16] Walter Rudin. Real and complex analysis. McGraw-Hill, New York, NY, 3rd edition, 1987.
[17] Raphaël Salem. On singular monotonic functions whose spectrum has a given Hausdorff dimension. Ark. Mat., 1:353–365, 1951.
[18] Elias Stein. Harmonic Analysis Real-Variable Methods, Orthogonality, and Oscillatory Integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1993.
[19] John Thorpe. Elementary Topics in Differential Geometry. Undergraduate Texts in Mathematics. Springer, New York, NY, 1979.
[20] Vitaly Ushakov. Developable surfaces in Euclidean space. J. Austral. Math. Soc. Ser. A, 66(3):388–402, 1999.
[21] Junjie Zhu. Fourier dimension of conical and cylindrical hypersurfaces, 2024.

Department of Mathematics, 1984 Mathematics Road, University of British Columbia, Vancouver, BC Canada V6T 1Z2

E-mail address: [email protected]

Fourier dimension of constant rank hypersurfaces

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

1.1. Constant rank hypersurfaces

1.2. Main Result

Theorem 1.1.

Remark.

1.3. Overview of the proof

1.4. Proof of the main result and organization of the manuscript

Theorem A ([14]).

Proposition 1.2.

1.5. Notations

1.6. Acknowledgments

2. Upper bounds on the Fourier dimension: Examples

2.1. Tangent surface generated by a Helix

Proposition 2.1.

Proof.

Lemma 2.2.

Lemma 2.3.

Proof of Lemma 2.2.

Proof of Lemma 2.3.

2.2. A tangent surface generated by a perturbed helix

Proposition 2.4.

Lemma 2.5.

Proof of Proposition 2.4 assuming Lemma 2.5.

Lemma 2.6.

Lemma 2.7.

Proof of Lemma 2.6.

Proof of Lemma 2.7.

Proof of Lemma 2.5.

Remark.

3. Proof of Proposition 1.2

Definition 3.1.

Lemma B.

Remark.

3.1. The Main Proof

Lemma 3.2.

Proof of Proposition 1.2 assuming Lemma 3.2.

Lemma 3.3.

Lemma 3.4.

Proof of Lemma 3.3.

Proof of Lemma 3.4.

3.2. Existence of TsT_{s}

Proof of Lemma 3.2.

Remark.

4. Appendix

4.1. Hausdorff and Fourier dimensions

4.2. Fourier dimension of the Cartesian product

Lemma 4.1.

Proof.

4.3. Reduction to compactly supported measure

Lemma 4.2.

Proof.

4.4. Morse’s lemma

Lemma 4.3.

Remark.

Proof.

4.5. Oscillatory Integrals

Proposition 4.4.

Proof.

Lemma 4.5.

Proof.

Theorem 4.6.

Proof.

References

3.2. Existence of $T_{s}$