Packing sets in Euclidean space by affine transformations

Let $\mathcal{A}$ be a metric space of maps $f:\mathbb{R}^{d}\to\mathbb{R}^{d}$, $d\geq 1$.

For Borel sets $E\subset\mathbb{R}^{d}$, $\Lambda\subset\mathcal{A}$, we define

The packing problem in this setting can be formulated as follows: Is it possible for a set of zero $d-$dimensional Lebesgue measure to contain an image $f(E)$ of $E$ for every $f\in\Lambda$?

The study of the packing problem has a long history, see for example [14], [31], and [23]. For the planar case, lines and circles are special cases of curve-packing problems. If we consider $E$ as a line segment in the plane, and let $\mathcal{A}$ be the set of all rigid motions, then the above problem dates back to the works of Besicovitch [1], [2] in the 1920s. In these papers, he constructed a set named after him which has zero Lebesgue measure and contains a line segment of unit length in every direction. Now let $E=S^{1}$, the unit circle, let $\mathcal{A}=\mathbb{R}^{2}\times\mathbb{R}_{>0}$, and define for each $\gamma=(z,r)\in\mathcal{A}$, $x\in\mathbb{R}^{2}$,

Then $\Lambda(S^{1})$ is the union of all circles centered at $z$, of radius $r$, for $(z,r)\in\Lambda$. In 1968, Besicovitch and Rado [3], and Kinney [25] showed that there exists a set in the plane of Lebesgue measure zero containing a circle of radius $r$ for each $r>0$ (see also Davies [9]). Later, in 1980, Talagrand [44] proved that there exists a set of measure zero containing a circle centered at $x$, for all $x$ on a given straight line. Thus it is natural to ask which conditions will guarantee that the union of lines (circles) has positive measure.

In this paper, we will study the following question, which generalizes the question for lines and circles above: Under which conditions on the dimensions of $E$ and $\Lambda$, the Lebesgue measure of $\Lambda(E)$ is positive? Note that when this holds, we certainly can not pack $\Lambda(E)$ into any set of zero Lebesgue measures. Similarly, we can ask: Given a constant $0<u\leq d$, how large do the Hausdorff dimensions $\dim_{\mathcal{H}}E$ and $\dim_{\mathcal{H}}\Lambda$ need to be to ensure that

holds?

For circles in the plane, in 1985, Bourgain [4], and independently Marstrand [28], demonstrated that if the centers form a set of positive Lebesgue measure, then the union of the circles must have positive Lebesgue measure, which answer a question by Falconer [14] (see also Bourgain [5]). Earlier Stein [43], as a consequence of his work on the spherical means maximal operator, proved that the same conclusion holds true for spheres in $\mathbb{R}^{d},d\geq 3$. The case $d=2$ turned out to be much more difficult. Since the spheres have dimension $d-1$, we can expect that it is enough for the centers of the spheres to form a set of Hausdorff dimension bigger than one to give a positive result. This was shown by Mitsis [34] for $d\geq 3$ and by Wolff [49] for $d=2$ (see also D. Oberlin [35]). In a paper published in 1997, Wolff [47] also showed that if the set of centers has Hausdorff dimension $s$, $0<s\leq 1$, the corresponding union of circles has dimension of at least $1+s$. For results in higher dimensions, see D. Oberlin [36].

We can get similar results by replacing the circle in the plane with rather general smooth curves in $\mathbb{R}^{d}$. If $E$ is a nondegenerate (i.e., derivatives span the whole space) curve in $\mathbb{R}^{d}$, $d\geq 3$, Ham, Ko, Lee, and Oh [17] showed that if the set of translations and dilations has dimension $>\alpha$, for $0<\alpha\leq d-1$, then the union of all curves has dimension $\geq\alpha+1$. For results in the plane, see [22]. Simon and Taylor [41], [42] studied properties of $A+\Gamma$, where $\Gamma$ is a planar curve with at least one point of non-vanishing curvature. For general results on the packing of curves, surfaces, and manifolds, see for example Falconer [13], Wisewell [46].

Turning to the case of affine hyperplanes in $\mathbb{R}^{d}$, we denote the Grassmanian manifold of all affine hyperplanes by $\mathcal{G}(d,d-1)$. Then if the set of hyperplanes has Hausdorff dimension larger than $1$, the union of these hyperplanes has a positive Lebesgue measure. In fact, we can say more, if $0<s<1$, and the dimension of the set of hyperplanes is $s$, then the union of hyperplanes will have dimension at least $d-1+s$. These results are due to D. Oberlin [36]. Later, in [37], he generalized to affine $k$-planes in $\mathbb{R}^{d}$. More precisely, he showed that if the set of affine $k$-planes has Hausdorff dimension $\alpha>(k+1)(d-k)-k$, then the union of corresponding $k$-planes has a positive Lebesgue measure. The result is sharp, in the sense that for every $\varepsilon>0$, there exists a set of $k$-planes of dimension $(k+1)(d-k)-k-\varepsilon$ such that the union of these $k-$planes has zero Lebesgue measure. In 2016, R. Oberlin [38] conjectured that if a set of lines $L$ in $\mathbb{R}^{d}$ has dimension $\geq 2(k-1)+\beta$, with integer $1\leq k\leq d$, then the union of these lines has dimension at least $k+\beta$. Recently, this conjecture was proved by Zahl [50]. For the variant of this problem when $L$ is replaced by a set of $k-$dimensional affine subspaces in $\mathbb{R}^{d}$, see [15], [20], [19], and [16]. For results on packing skeletons of polytopes, see [24], [45], and [7].

In this paper, instead of limiting the study to certain smooth objects, we will consider the packing problem for general Borel sets in $\mathbb{R}^{d}$. In particular, we are interested in packing problems in the case where $\mathcal{A}$ is the set of dilations and translations; rigid transformations; or similarity transformations.

Similar questions for rigid motions in finite field settings have also been studied in [39].

In this section, we will discuss the results of packing problems by using dilations and translations.

Let $d\geq 1$, we denote $\Gamma(d)=\mathbb{R}^{d}\times\mathbb{R}_{>0}$ as the set of all translations and dilations. For each $\gamma=(z,r)\in\Gamma(d)$ and $x\in\mathbb{R}^{d}$, we define $\gamma(x)=rx+z$. For $E\subset\mathbb{R}^{d}$, and $\Gamma\subset\Gamma(d)$, set

In other words, $\Gamma(E)$ is the union of all dilated and translated copies $\gamma(E)$ of $E$, with $\gamma\in\Gamma$. If $\Gamma$ is of the form $Z\times T$, where $Z\subset\mathbb{R}^{d}$, $T\subset\mathbb{R}_{>0}$, by denoting $TE=\{rx:x\in E,r\in T\}$, we can write

We restate the packing problem using dilations and translations as follows: For Borel sets $E$ and $\Gamma$, can a set of zero $d-$dimensional Lebesgure measure contains dilated and translated copies $\gamma(E)=rE+z$ of $E$ for each element $\gamma=(z,r)\in\Gamma$?

Recently, by assuming that the set of dilations and translations has product form $Z\times T\subset\mathbb{R}^{d}\times\mathbb{R}_{>0}$, Hambrook and Taylor [18] proved the following theorem. Throughout this paper, $\dim_{\mathcal{F}}$ denotes the Fourier dimension, see section 2 for details.

Theorem A holds in the more general form as follows. Throughout this paper, $\dim_{\mathcal{S}}$ denotes the Sobolev dimension. We say that a set $A\subset\mathbb{R}^{d}$ has Sobolev dimension $\dim_{\mathcal{S}}A\geq s$ if $A$ carries a Borel probability measure $\mu$ such that

The proof of Theorem B is as in [18] (see Section 2 for details).

While Theorem A is sharp (for example, take $E=\{0\}\subset\mathbb{R}^{d}$, $T=\{1\}$ and $Z\subset\mathbb{R}^{d}$ such that $\dim_{H}Z=d$, while $\mathcal{L}^{d}(Z)=0$), the role of the Hausdorff dimension of $E$ in this problem is not provided. Additionally, finding lower bounds for $\dim_{\mathcal{F}}(TE)$ in Theorem A is important.

Our first main result is the following, which answers the above questions.

To emphasize the importance of the estimates for lower bounds of the Fourier dimension, we give an immediate corollary of Theorem 1.1 to the $k$-fold sum-product set without proof, see [31, Proposition 3.14]. Let $A\subset\mathbb{R}^{d}$, we denote $A^{(k)}=\underbrace{A+\cdots+A}_{k}$ as the $k-$fold sum-set of $A$.

We also obtain a similar result when $\Gamma$ is a general set in $\Gamma(d)$.

More generally, we also consider the packing problem by multi-parameter dilations and translations as follows. Denote $\tilde{\Gamma}(d)=\mathbb{R}^{d}\times\mathbb{R}_{>0}^{d}$ as the set of all multi-parameter dilations and translations. For each $\tilde{\gamma}=(z,r)=(z_{1},\dots,z_{d},r_{1},\dots,r_{d})\in\tilde{\Gamma}(d)$, and for each $x\in\mathbb{R}^{d}$ we define

Let $E\subset\mathbb{R}^{d}$, and $\tilde{\Gamma}\subset\tilde{\Gamma}(d)$ be Borel sets, set

Our next main result is the following.

We shall now investigate the packing problem in Euclidean space by using another class of affine transformations, the set of all rigid motions.

Let $d\geq 2$, we denote $E(d)=O(d)\times\mathbb{R}^{d}$ as the Euclidean group or the set of all rigid motions. For each $\theta=(g,z)\in E(d)$, $x\in\mathbb{R}^{d}$, we define $\theta(x)=g(x)+z$. Let $\Theta\subset E(d)$ and $E\subset\mathbb{R}^{d}$, we define

When $\Theta=G\times Z$, where $G\subset O(d)$ and $Z\subset\mathbb{R}^{d}$, we denote

From definition, $\Theta(E)$ is the union of all rotated and translated copies $\theta(E)=g(E)+z$ of $E$, for $\theta=(g,z)\in\Theta$. We will find conditions on the dimensions of $E$ and $\Theta$ to ensure that the set containing all rotated and translated copies $\theta(E)=g(E)+z$ of $E$ for each element $\theta=(g,z)\in\Theta$ has positive Lebesgue measure.

Before stating the theorem, we will recall the definition of spherical Fourier dimension (see [29]). For a given Borel set $E\subset\mathbb{R}^{d}$, the spherical Fourier dimension of $E$, denoted by $\dim_{\mathcal{SF}}E$, is defined by

where $\sigma(\mu)(r)$ is the $L^{2}$ spherical average of the Fourier transform of $\mu$, see (2.3). We have the following lemma, see Section 2 for more details.

The main result in this section is the following.

Recalling Lemma 1.1 we have

Based on the above theorem, it is plausible to make the following conjecture for packing arbitrary Borel sets in Euclidean space using rigid transformations.

Instead of using rigid motions, we can extend the study of the packing problems to the class of similarity transformations.

Let $d\geq 2$, for each $\omega=(g,z,r)\in\Omega(d)=O(d)\times\mathbb{R}^{d}\times\mathbb{R}_{>0}$, the set of similarity transformations, and $x\in\mathbb{R}^{d}$, we define $\omega(x)=rg(x)+z$. Let $\Omega\subset\Omega(d)$ and $E\subset\mathbb{R}^{d}$, we set

When $\Omega=G\times Z\times T$, where $G\subset O(d)$, $Z\subset\mathbb{R}^{d}$, and $T\subset\mathbb{R}_{>0}$, we denote

Observe that $\Omega(E)$ is the union of all dilated, rotated, and translated copies $\omega(E)$ of $E$, for each $\omega\in\Omega$.

Similar to the packing problem using rigid motions, we ask: Under which conditions of the dimensions of $E$ and $\Omega$, we can not pack $\Omega(E)$ into a set of zero Lebesgue measure in $\mathbb{R}^{d}$?

The main result in this section is the following.

In this section, we will discuss analogous packing results in vector spaces over finite fields.

Let $\mathbb{F}_{q}^{d}$ be the $d-$dimensional vector space over a finite field $\mathbb{F}_{q}$ with $q$ elements, where $q$ is an odd prime power, $d\geq 2$.

For each $x\in\mathbb{F}_{q}^{d},\theta=(g,z)\in O(d)\times\mathbb{F}_{q}^{d}$, we define $\theta(x)=gx+z$. For $E\subset\mathbb{F}_{q}^{d}$, and $\Theta\subset O(d)\times\mathbb{F}_{q}^{d}$, we define

The following three theorems are proved in [39] using bounds on the incidence between points and rigid motions. We refer the reader to [39] for a discussion on the sharpness of these results.

We provide simple alternative proofs of these results using some of the ideas from our ${\mathbb{R}}^{d}$ results. However, we do not currently have an ${\mathbb{R}}^{2}$ variant of Theorem 1.6. We hope to address this issue in the sequel.

The structure of the rest of the paper is as follows: In Section 2, we will give some notations, preliminaries, and lemmas needed for the rest of the paper. The proofs of Theorems 1.1, 1.4 and 1.5 will be given in Section 3. Theorems 1.2 and 1.3 will be proved in Section 4. In Section 5, we will review some basic background on Fourier analysis over finite fields, and then give the proof of Theorem 1.6. Some examples related to our results will be presented in Section 6.

In particular, Frostman’s lemma implies that given any exponent $0<s<\dim_{\mathcal{H}}(E)$, there exists a probability measure $\mu$ on $E$ such that

A measure satisfying condition (2.1) is often called an $s$-dimensional Frostman measure (or $s$-Frostman measure). The $s$-energy integral of a measure $\mu\in\mathcal{M}(\mathbb{R}^{d})$ (see [30, 31]) is

If $\mu\in\mathcal{M}(\mathbb{R}^{d})$ satisfies the Frostman condition $\eqref{eq_Frostman_measure}$, then $I_{t}(\mu)<\infty$ for all $0<t<s$.

We have for any Borel set $A\subset\mathbb{R}^{d}$ with Hausdorff dimension $\dim_{\mathcal{H}}A>0$, (see Theorem 8.9 in [30]),

The Fourier dimension of a set $A\subset\mathbb{R}^{d}$ is

The Sobolev dimension of a measure $\mu\in\mathcal{M}(\mathbb{R}^{d})$ is

We will say that a set $A\subset\mathbb{R}^{d}$ has Sobolev dimension $\dim_{\mathcal{S}}A\geq s$ if $A$ carries a Borel probability measure $\mu$ such that $\dim_{\mathcal{S}}\mu\geq s$. The greater the Sobolev dimension is, the smoother the measure is in some sense. We recall the following well-known result, see [31].

Let $\mu\in\mathcal{M}(\mathbb{R}^{d})$ be an $s$-Frostman measure. We have the following ball average estimate (see [31, Section 3.8]),

Next, given a Radon measure $\mu$ with compact support on $\mathbb{R}^{d}$, $d\geq 2$, we define the $L^{2}$ spherical averages of the Fourier transform of $\mu$ by

where $\sigma$ is the surface measure on the unit sphere $S^{d-1}$. In [29], Mattila developed a method to study Falconer’s distance problem by studying the decay rates of spherical averages of fractal measures, namely, the supremum of the numbers $\beta$ for which for all $r>1$, one has

For any $s$-Frostman measure $\mu$ we have for all $\varepsilon>0$, $r>1$,

The first two estimates were proved in [29]. The last estimate is the deepest. It is due to Wolff [48] for $d=2$, to Du-Guth-Ou-Wang-Wilson-Zhang [10] for $d=3,3/2<s\leq 2$, and to Du-Zhang [11] for $d=3,2\leq s<3$, and $d\geq 4$. The first estimate is always sharp and for $d=2$ they all are sharp.

The lower bounds for the spherical Fourier dimension in Lemma 1.1 follow from the best known decay of spherical averages (2.5).

In this section, we will recall some results needed for the proof of the theorems.

First, we give a proof for Theorem B.