New improvement to Falconer distance set problem in higher dimensions

We will extend and refine the good tube/bad tube and decoupling method pioneered by [10] for dimension $d=2$ and continued in [5] for even dimensions $d$. To state our contributions, we find it helpful to first recall the good tube/bad tube decomposition in [10], [5]. Fix $\alpha>\frac{d}{2}$ and a set $E$ with $\dim_{H}(E)>\alpha$; thus, we can find two $\alpha$-dimensional Frostman measures $\mu_{1},\mu_{2}$ supported on separated subsets of $E$. Then we fix a single scale $r$, and consider the set of $r$-tubes (i.e. $r$-neighborhoods of lines) that intersect $E$. We say that an $r$-tube $T$ is good if $\mu_{2}(T)\lesssim r^{\frac{d}{2}-\varepsilon}$, and bad otherwise. The paper [10] developed a decoupling framework that can handle good tubes very well, so it remains to show that there are very few bad tubes. When $d=2$, this was solved by appealing to the $n=2$ case of Orponen’s radial projection theorem, which roughly states that given a Frostman measure $\mu$ in $\mathbb{R}^{n}$ with dimension $>n-1$, most tubes satisfy $\mu_{2}(T)\lesssim r^{n-1}$. When $d>2$ is even, we can no longer directly apply the radial projection theorem because $\alpha$ is usually $\leq d-1$; to overcome this [5] came up with a partial fix by first projecting $\mu_{2}$ onto a generic $(\frac{d}{2}+1)$-dimensional subspace of $\mathbb{R}^{d}$ (assuming $d$ is even), so the projected measure $\mu_{2*}$ still remains $\alpha$-dimensional, and then applying Orponen’s radial projection theorem to $\mu_{2*}$ in $\mathbb{R}^{\frac{d}{2}+1}$. Despite the seemingly wasteful application of the initial orthogonal projection, the threshold $r^{\frac{d}{2}-\varepsilon}$ is sharp up to $\varepsilon$: the example of a $(\frac{d}{2}+1)$-hyperplane shows that $r^{\frac{d}{2}}$ is the best threshold one can get in general (and we get an even worse threshold $r^{\frac{d-1}{2}}$ for odd dimensions via an $(\frac{d+1}{2})$-hyperplane example; see e.g. Section 5.2 of [5]).

To improve upon [5], we must find a way to deal with this hyperplane example. As it so happens, we are elated if this situation occurs: if our set $E$ is contained in a lower-dimensional hyperplane $H=\mathbb{R}^{n}$, then we can simply work in $\mathbb{R}^{n}$ where existing distance set results are already better than what we need. Thus, it would be nice to establish a dichotomy: either we can improve upon the barrier $r^{\lfloor\frac{d}{2}\rfloor}$, or it turns out that much of $E$ is contained in a lower-dimensional hyperplane. Fortunately, this wish is indeed true and is one of our main innovations of the paper. More precisely, we establish a framework, for the first time in the context of the distance problems, that analyzes how much a fractal measure concentrates near a low dimensional plane hence allows us to reduce the problem to the aforementioned dichotomy. We show that either (1) the measure is dictated by wave packets (i.e. microlocalized pieces of the measure) that display good transversality. In this case, desirable multilinear estimates or projection results can apply to deduce an improved threshold $r^{\alpha-\epsilon}$ (a significant improvement over the barrier $r^{\lfloor\frac{d}{2}\rfloor}$ and would lead to gains in the decoupling step), or (2) the problem can be reduced to lower dimensions hence existing distance set results can apply.

On the technical level, this is achieved by a careful analysis of the interaction between the plates of intermediate dimensions (i.e. neighborhood of planes) and wave packets. Our starting point is a new radial projection theorem of the third author [22], which we precisely state as Theorem 3.1. Roughly, it states that most $r$-tubes either have $\mu_{2}$-mass $\lesssim r^{\alpha-\varepsilon}$ or lie in some heavy $(r^{\kappa},\lceil\alpha\rceil)$-plate, which is the $r^{\kappa}$-neighborhood of a $\lceil\alpha\rceil$-dimensional hyperplane with $(\mu_{1}+\mu_{2})$-mass $\gtrsim r^{\eta}$ (here, $\eta\ll\kappa\ll\varepsilon$). The main obstacle that we need to deal with then becomes the heavy $(r^{\kappa},\lceil\alpha\rceil)$-plates, and a key novelty of this paper is an upgrade of the good/bad framework that analyzes the interaction between these plates and the wave packets.

Note that this strategy in fact indicates a key difference between the distance problem in two dimensions and that in higher dimensions: in dimension three and higher, it is possible to study the distance set using information from the distance problem in lower dimensions. This is apparently not possible in the plane, in which case the measure is assumed to have Hausdorff dimension greater than one hence cannot concentrate on a line. As far as we are aware of, it seems that such a special feature of the distance problem in the higher dimensional setting has not been explored in the literature before.

Here are some more details, where we fix $d=3$ for concreteness. Following the setup in [5], we let $\mu_{1},\mu_{2}$ be $\alpha$-dimensional Frostman measures supported on subsets $E_{1},E_{2}\subset E$ with $\text{dist}(E_{1},E_{2})\gtrsim 1$. Let $R_{0}$ be a large number and define the scales $R_{j}=2^{j}R_{0}$. Fix a scale $j$; we shall work with $R_{j}^{-1/2+\beta}$-tubes. Let $G(x)$ to be the set of $y\in E_{1}$ such that $x,y$ don’t both lie in some heavy $(R_{j}^{-\kappa},2)$-plate. Define a good $R_{j}^{-1/2+\beta}$-tube to be one with mass $\lesssim R_{j}^{-\alpha/2+\varepsilon}$ and doesn’t lie in any heavy plate, and $\mu_{1,g}$ to be a part of $\mu_{1}$ from all good tubes. With these definitions, we can try to follow the framework of [5] and show that $\mu_{1}|_{G(x)}$ and $\mu_{1,g}$ are close in $L^{1}$-norm. There are two subtleties with this approach. First, the error in $\|\mu_{1}|_{G(x)}-\mu_{1,g}\|_{L^{1}}$ may contain contributions from tubes $T$ at the border of $G(x)$, i.e. $T\subset G(x)$ but $2T\not\subset G(x)$. To overcome this issue, we introduce a probabilistic wiggle. If we work with heavy $(aR_{j}^{-\kappa},2)$-plates for a uniformly random $a\in[1,2]$, then the borderline error will be small on average, so there exists a choice for $a$ to make the borderline error small.

The second issue is that we do not have the luxury of working at a single scale, so we need to introduce some ideas from [25, Appendix B]. Specifically, we will construct a decreasing sequence $\mathbb{R}^{3}\supset G_{0}(x)\supset G_{1}(x)\supset\cdots$ such that $\mathbb{R}^{3}\setminus G_{0}(x)$ is contained in a $(R_{0}^{-\kappa/2},2)$-plate, $\mu(G_{0}(x)\setminus G_{\infty}(x))\leq R_{0}^{-\kappa/2}$, and $G_{j}(x)$ is disjoint from all heavy $(aR_{j}^{-\kappa},2)$-plates containing $x$ (where $a\in[1,2]$ is the probabilistic wiggle). Since all the $G_{j}$’s are close in measure, we can combine multiscale information to show that $\mu_{1}|_{G_{0}(x)}$ and $\mu_{1,g}$ are close in $L^{1}$-norm.

The final step is to show $G_{0}(x)$ is large. This is equivalent to showing that any $(R_{0}^{-\kappa/2},2)$-plate has small ($\leq\frac{1}{2}$) measure for $R_{0}$ large enough. If this is not true, then by compactness we can find a hyperplane with nonzero measure, and then we reduce to Falconer in 2D. In this case, we have strong bounds for distance sets (e.g. see Wolff [29]) because $\dim_{H}(E)>1.5$ is large.

We point out that, in some sense, the dimensional threshold we obtained already fully exploited the power of the methods of this paper and the companion paper [6]. The radial projection theorem of [22] is sharp in the following sense: let $E_{1},E_{2}$ each be unions of $r^{-\alpha}$ many $r$-balls satisfying an $\alpha$-dimensional spacing condition such that $\mathrm{dist}(E_{1},E_{2})\geq\frac{1}{2}$, and let $\mu_{1},\mu_{2}$ be probability measures supported on $E_{1},E_{2}$ respectively such that each $r$-ball in $E_{i}$ has $\mu_{i}$-measure $r^{\alpha}$, $i=1,2$. Then, we know that many $r$-tubes can intersect at least one $r$-ball in $E_{1}$ and one $r$-ball in $E_{2}$. Thus, many $r$-tubes can have both $\mu_{1}$ and $\mu_{2}$-mass at least $r^{\alpha}$, so $r^{\alpha}$ is the best possible threshold for the good tubes in this paper. To make further progress, we suggest to look at improving the decoupling framework.

On the other hand, the threshold $r^{\alpha}$ is already stronger than all the previously best known thresholds for good tubes in all dimensions $d\geq 2$ (see [10, 5]). For comparison, in two dimensions, it was shown that one can reduce to good $r$-tubes whose measure is at most $\sim r$, which is weaker than $r^{\alpha}$. The possibility of obtaining such a better threshold for the good tubes underscores another key difference between the distance problem in higher dimensions and that in the plane. Indeed, one can easily see that for an evenly distributed probablity measure in the plane, the threshold $r$ would be the best possible.

In Section 2, we outline two main estimates and prove Theorem 1.2 using them. In Section 3, we list several results that we will use to bound the bad part, including the new radial projection estimate by the third author [22] and two geometric lemmas governing the physical location of small plates with large mass. In Section 4, we construct the good measure and prove the first main estimate - Proposition 2.1. In Section 5, we prove the second main estimate - Proposition 2.2 using refined decoupling. In Section 6, we prove Theorem 1.3 using the two main estimates and a framework of Liu [15]. In Section 7, we give some remarks about the extension of Theorem 1.2 to more general norms and its connection with the Erdős distance problem.

Throughout the article, we write $A\lesssim B$ if $A\leq CB$ for some absolute constant $C$; $A\sim B$ if $A\lesssim B$ and $B\lesssim A$; $A\lesssim_{\varepsilon}B$ if $A\leq C_{\varepsilon}B$; $A\lessapprox B$ if $A\leq C_{\varepsilon}R^{\varepsilon}B$ for any $\varepsilon>0$, $R>1$.

For a large parameter $R$, ${\rm RapDec}(R)$ denotes those quantities that are bounded by a huge (absolute) negative power of $R$, i.e. ${\rm RapDec}(R)\leq C_{N}R^{-N}$ for arbitrarily large $N>0$. Such quantities are negligible in our argument.

For $x\in\mathbb{R}^{d}$ and $t>0$, $B(x,t)$ is the ball in $\mathbb{R}^{d}$ of radius $t$ centered at $x$. A $\delta$-tube is the intersection of an infinite cylinder of radius $\delta$ and $B(0,10)$. This is not standard (usually, a tube is defined to be a finite $\delta$-cylinder with length $1$), but this definition won’t cause problems and is slightly more convenient for us.

For a set $E\subset\mathbb{R}^{d}$, let $E^{c}=\mathbb{R}^{d}\setminus E$, and $E^{(\delta)}$ the $\delta$-neighborhood of $E$. For subsets $E_{1},E_{2}\subset\mathbb{R}^{d}$, $\mathrm{dist}(E_{1},E_{2})$ is their Euclidean distance.

For $A\subset X\times Y$ and $x\in X$, define the slice $A|_{x}=\{y\in Y:(x,y)\in A\}$. Similar definition for $A|_{y}$, when $y\in Y$.

For a measure $\mu$ and a measurable set $G$, define the restricted measure $\mu|_{G}$ by $\mu|_{G}(A)=\mu(G\cap A)$ for all measurable $A\subset\mathbb{R}^{d}$.

We say a measure $\mu$ supported in $\mathbb{R}^{d}$ is an $\alpha$-dimensional measure with constant $C_{\mu}$ if it is a probability measure satisfying that

An $(r,k)$-plate $H$ in $\mathbb{R}^{d}$ is the $r$-neighborhood of a $k$-dimensional affine plane in the ball $B^{d}(0,10)$. More precisely,

where $P_{H}$ is a $k$-dimensional affine plane, which is called the central plane of $H$. We can also write $H=P_{H}^{(r)}\cap B(0,10)$. A $C$-scaling of $H$ is

Denote the surface of $H$ by

Since a $\delta$-tube is a $(\delta,1)$-plate, the same conventions apply to tubes.

To prevent the reader from feeling too attached to $B(0,10)$, we will see in the proofs that $B(0,10)$ can be replaced by any smooth bounded convex body that contains $B(0,10)$.

We say that an $(r,k)$-plate $H$ is $\gamma$-concentrated on $\mu$ if $\mu(H)\geq\gamma$.

Given a collection $\mathcal{H}$ of plates in $\mathbb{R}^{d}$ and a point $x\in\mathbb{R}^{d}$, define $\mathcal{H}(x):=\{H\in\mathcal{H}:x\in aH\}$, where $a$ is a “master parameter” that will be defined later.

We will work with a collection $\mathcal{E}_{r,k}$ of essentially distinct $(r,k)$-plates with the following properties:

• •

  Each $(\frac{r}{2},k)$-plate intersecting $B(0,1)$ lies in at least one plate of $\mathcal{E}_{r,k}$;

• •

  For $s\geq r$, every $(s,k)$-plate contains $\lesssim_{k,d}\left(\frac{s}{r}\right)^{(k+1)(d-k)}$ many $(r,k)$-plates of $\mathcal{E}_{r,k}$.

For example, when $k=1$ and $d=2$, we can simply pick $\sim r^{-1}$ many $r$-tubes in each of an $r$-net of directions. This generalizes to higher $k$ and $d$ via a standard $r$-net argument, which can be found in [22, Section 2.2].

We follow the first part of [5, Section 3.1]. Let $R_{0}$ be a large power of $2$ that will be determined later, and let $R_{j}=2^{j}R_{0}$. Construct a partition of unity

where $\psi_{0}$ is supported in the ball $|\omega|\leq 2R_{0}$ and each $\psi_{j}$ for $j\geq 1$ is supported on the annulus $R_{j-1}\leq|\omega|\leq R_{j+1}$. Importantly, we may choose $\psi_{j}$ such that $\|\check{\psi}_{j}\|_{L^{1}}\leq C$ for some absolute constant $C$ and all $j\geq 1$. For example, choose $\psi_{j}$ to be Littlewood-Paley functions $\chi(x/R_{j})-\chi(x/R_{j-1})$, where $\chi$ is a smooth bump function that is $1$ on $B(0,1)$ and $0$ outside $B(0,2)$.

In $\mathbb{R}^{d}$, cover the annulus $R_{j-1}\leq|\omega|\leq R_{j+1}$ by rectangular blocks $\tau$ of dimensions approximately $R_{j}^{1/2}\times\cdots\times R_{j}^{1/2}\times R_{j}$, with the long direction of each block $\tau$ being the radial direction. Choose a smooth “partition of unity” with respect to this cover such that

The functions $\psi_{j,\tau}$ satisfy the following properties:

• •

  $\psi_{j,\tau}$ is supported on $\tau$ and $\|\psi_{j,\tau}\|_{L^{\infty}}\leq 1$;

• •

  $\check{\psi}_{j,\tau}$ is essentially supported on a $R_{j}^{-1/2}\times\cdots\times R_{j}^{-1/2}\times R_{j}^{-1}$ box $K$ centered at $0$, in the sense that $|\check{\psi}_{j,\tau}(x)|\leq\mathrm{RapDec}(R_{j})$ if $\mathrm{dist}(x,K)\gtrsim R_{j}^{-1/2+\beta}$ (and the implicit constant in the decay $C_{N}R_{j}^{-N}$ is universal only depending on $N$);

• •

  $\|\check{\psi}_{j,\tau}\|_{L^{1}}\lesssim 1$ (the implicit constant is universal).

For each $(j,\tau)$, cover the unit ball in $\mathbb{R}^{d}$ with tubes $T$ of dimensions approximately $R_{j}^{-1/2+\beta}\times\cdots\times R_{j}^{-1/2+\beta}\times 20$ with the long axis parallel to the long axis of $\tau$. The covering has uniformly bounded overlap, each $T$ intersects at most $C(d)$ other tubes. We denote the collection of all these tubes as $\mathbb{T}_{j,\tau}$. Let $\eta_{T}$ be a smooth partition of unity subordinate to this covering, so that for each choice of $j$ and $\tau$, $\sum_{T\in\mathbb{T}_{j,\tau}}\eta_{T}$ is equal to 1 on the ball of radius 10 and each $\eta_{T}$ is smooth.

For each $T\in\mathbb{T}_{j,\tau}$, define an operator

which, morally speaking, maps $f$ to the part of it that has Fourier support in $\tau$ and physical support in $T$. Define also $M_{0}f:=(\psi_{0}\hat{f})^{\vee}$. We denote $\mathbb{T}_{j}=\cup_{\tau}\mathbb{T}_{j,\tau}$ and $\mathbb{T}=\cup_{j\geq 1}\mathbb{T}_{j}$. Hence, for any $L^{1}$ function $f$ supported on the unit ball, one has the decomposition