Discretized Radial Projections in $\R^{d}$

Radial projections have also been used to study the Falconer distance set problem, which asks for lower bounds on the Hausdorff dimension of the distance set $\Delta(X):=\{|x-y|:x,y\in X\}$ given the value of $\dim_{H}(X)$ for some $X\in\R^{d}$. In two dimensions, Wolff [33] used Fourier analysis to show that if $\dim_{H}(X)\geq\frac{4}{3}$, then $\Delta(X)$ has positive Lebesgue measure. Using Orponen’s radial projection theorem [17], Guth-Iosevich-Ou-Wang [7] used a good-bad tube decomposition and decoupling to improve the threshold to $\dim_{H}(X)\geq\frac{5}{4}$. See also works of Keleti-Shmerkin [14] [14], Shmerkin [28], Liu [15], and Stull [29] which provide better lower bounds for $\dim_{H}(\Delta(X))$ given that $\dim_{H}(X)\in(1,\frac{5}{4})$. In higher dimensions, the works of Du-Iosevich-Ou-Wang-Zhang [3] and Wang-Zheng [32] used a good-bad tube decomposition using Orponen’s radial projection theorem and decoupling techniques [17] to provide state-of-the-art results when the dimension $d$ is even; when $d$ is odd, a more classical approach purely based on decoupling gave the best estimates [6], [9]. More recently, Shmerkin and Wang [27] prove a radial projection theorem in the spirit of this paper to provide an improved lower bound when $\dim_{H}(X)=\frac{d}{2}$, $d=2,3$; using their framework combined with updated results of [19], one can show for example that $\dim_{H}(\Delta(X))\geq\frac{5}{8}$ when $X\subset\R^{3}$ satisfies $\dim_{H}(X)=\frac{3}{2}$. In fact, all of these works prove lower bounds on the size of the pinned distance set, $\Delta_{x}(X):=\{|x-y|:y\in X\}$. In the forthcoming companion papers [4], [5], we use Theorem 1.1 to improve the lower bounds for the Falconer distance set problem in all dimensions $d$.

Very recently, radial projections in dimension $2$ have been used to prove the ABC sum-product conjecture and Furstenberg set conjecture, and yield progress on the discretized sum-product problem [21], [22]. It is natural to wonder whether the exciting progress in 2 dimensions will generalize to higher dimensions. The starting point of the breakthrough work of [21] (which was also used in [22]) is a sharp radial projection theorem in 2 dimensions, [19, Theorem 1.1]. We hope to use our higher dimensional radial projection theorem to prove analogous results to [21], [22] in all dimensions.

We deduce Theorem 1.1 from $\delta$-discretized versions. The following notation will be used throughout this paper.

A $(\delta,s,C,k)$-set of balls cannot be concentrated in a small neighborhood of a $k$-plane, while a $(\delta,s,C,k)$-set of tubes cannot be concentrated in a small neighborhood of a $(k+1)$-plane.

The main ingredient in the proof of Theorem 1.1 is an $\varepsilon$-improvement to the (dual) Furstenberg set problem that generalizes Theorem 1.3 in [18] to higher dimensions.

In fact, we can prove the following refined version of Theorem 1.8.

Using Theorem 1.11, a bootstrap argument based on [19] gives the following.

The main proof ideas for Theorem 1.8 are as follows:

1. 1.

   Perform a standard multiscale decomposition argument due to [18] to reduce the original problem to two building blocks: the case when $\mathcal{P}$ is a $(\delta,s)$-set and when $\mathcal{P}$ is a $t$-regular set. The first case doesn’t happen all the time and has no loss by an elementary incidence argument, so we focus on gaining an $\varepsilon$-improvement in the second case. A $t$-regular set $\mathcal{P}$ has the special property that $|\mathcal{P}\cap Q|$ is still a $(\Delta,t)$-set for $Q\in\mathcal{D}_{\Delta}(\mathcal{P})$, $\Delta=\delta^{1/2}$.

2. 2.

   If $\mathcal{P}$ is $t$-regular with $\Delta=\delta^{1/2}$, we may find a $\Delta$-tube $\mathbf{T}$ such that upon dilation of $\mathbf{T}$ to $[0,1]^{d}$, we obtain a new Furstenberg problem with the ball set having a quasi-product structure. See Appendix A of [18].

3. 3.

   Finally, we will use discretized sum-product type arguments to conclude an $\varepsilon$-improvement to the dual Furstenberg problem assuming $\mathcal{P}=X\times Y\subset\R^{d-1}\times\R$ has a quasi-product structure. In very rough terms, we shall lift $Y$ to have dimension close to $1$, and apply multi-linear Kakeya. This idea of lifting the dimension was found in He’s work on a higher-rank discretized sum-product theorem [11] in a slightly different context.

To prove Theorem 1.11, we use a similar multiscale decomposition argument as in (1) to reduce to two building blocks: a smaller version of the setting of Theorem 1.11 and a smaller version of Theorem 1.8. The smaller version of Theorem 1.11 has no loss by an elementary incidence argument, and the smaller version of Theorem 1.8 admits a gain.

For Theorem 1.13, we first prove the case when $\mu,\nu$ are supported in a $r_{0}K$ plate (where $K$ is a small power of $r_{0}^{-1}$). This uses a similar argument as in [19, Lemma 2.8]. The general case follows from applying this special case many times.

In Section 2, we introduce some key concepts that will be used throughout the paper. In Sections 3 through 5, we prove Theorem 1.8 first for quasi-product sets following ideas of [10], and then for regular sets and finally for general sets following [18]. In Section 6, we prove Theorem 1.11 from Theorem 1.8. In Section 7, we generalize a radial projection theorem of Shmerkin [25, Theorem 6.3] that enables us to assume our sets have power decay around $k$-planes. In Section 8, we prove Theorem 8.1 following ideas from [19]. Finally, in Section 9, we prove Theorem 1.1 and 1.2 from the discretized results.

Acknowledgments. The author is supported by a NSF GRFP fellowship. The author would like to thank Xiumin Du, Tuomas Orponen, Yumeng Ou, Pablo Shmerkin, Hong Wang, and Ruixiang Zhang for helpful discussions. We thank Paige Bright and Yuqiu Fu for suggesting to include a higher-dimensional version of Beck’s theorem in this paper.

We use $A\lesssim B$ to denote $A\leq CB$ for some constant $C$. We use $A\lesssim_{N}B$ to indicate the constant $C$ can depend on $N$. We will also use $A\lessapprox B$ in future proofs; its exact meaning will always be clarified when used.

For a finite set $A$, let $|A|$ denote the cardinality of $A$. If $A$ is infinite, let $|A|$ denote the Lebesgue measure of $A$.

For a set $A$, let $A^{c}=\R^{d}\setminus A$.

For a tube $T$, let $\ell(T)$ denote the central line segment of $T$.

For a set $E$, let $E^{(\delta)}$ be the $\delta$-neighborhood of $E$.

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

For a measure $\mu$ and a set $G$, define the restricted measure $\mu|_{G}$ by $\mu|_{G}(A)=\mu(G\cap A)$. The renormalized restricted measure is $\mu_{G}=\frac{1}{\mu(G)}\mu|_{G}$.

For vectors $v_{1},\cdots,v_{i}\in\R^{d}$, $1\leq i\leq d$, the quantity $|v_{1}\wedge\cdots\wedge v_{i}|$ is the non-negative volume of the parallelepiped spanned by $v_{1}$ through $v_{i}$.

$B(x,r)$ is the ball in $\R^{d}$ of radius $r$ centered at $x$. We also use the notation $B_{r}$ for an arbitrary $r$-ball in $\R^{d}$.

For sets $A,B$ and $P\subset A\times B$, let $A\overset{P}{+}B:=\{a+b:(a,b)\in P\}$.

We work in $\R^{d}$. An $(r,k)$-plate is the $r$-neighborhood of a $k$-dimensional hyperplane in $\R^{d}$. We construct a set $\mathcal{E}_{r,k}$ of $(r,k)$-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 $\lesssim\left(\frac{s}{r}\right)^{(k+1)(d-k)}$ 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, but we haven’t seen it in the literature, so we provide a precise construction.

An $r$-net of a metric space is a subset $S$ such that $B(x,r)\cap B(y,r)=\emptyset$ for $x\neq y,x,y\in S$. The affine Grassmanian manifold $\mathbb{A}(k,d)$ is the set of all $k$-planes in $\R^{d}$. By counting degrees of freedom, we see that $\dim\mathbb{A}(k,d)=(k+1)(d-k)$. Any such plane is uniquely $V=V_{0}+a$ for some $k$-dimensional subspace $V_{0}$ and $a\in V_{0}^{\perp}$. For $V=V_{0}+a$ and $W=W_{0}+b$, define their distance $d_{\mathbb{A}}$ to be (following Section 3.16 of [16]):

where $\pi_{V_{0}}:\R^{d}\to V_{0}$ and $\pi_{W_{0}}:\R^{d}\to W_{0}$ are orthogonal projections, and $\|\cdot\|_{op}$ is the usual operator norm for linear maps. Let $\mathbb{A}_{0}(k,d)$ be the submanifold of $k$-planes $V_{0}+a$ with $a\in B(0,10)$. Since the manifold $(\mathbb{A}_{0}(k,d),d_{\mathbb{A}})$ is compact and smooth, it can be covered by finitely many charts that are $\sim 1$-bilipschitz to a subset of $\R^{(k+1)(d-k)}$.

From a maximal $cr$-net $\mathcal{N}$ of the set of affine planes of $\mathbb{A}_{0}(k,d)$ with $c>0$ a sufficiently small constant, we can construct a set $\mathcal{E}_{r,k}$ of $(r,k)$-plates whose central planes are the elements of $\mathcal{N}$. We now check the two properties for $\mathcal{E}_{r,k}$.

To prove the first property, let $H$ be a $(\frac{r}{2},k)$-plate intersecting $B(0,1)$. Then the central plane $P=P_{H}$ must lie at distance $\leq 2cr$ from some element $Q$ of $\mathcal{N}$ (otherwise, we can add it to the net). Let $P=P_{0}+a$ and $Q=Q_{0}+b$. Hence, $\|\pi_{P_{0}}-\pi_{Q_{0}}\|_{op}\leq 2cr$ and $|a-b|\leq 2cr$, so for $x\in P\cap B(0,10)$ (so $x-a\in P_{0}$),

Now, note that $\pi_{Q_{0}}(x-a)+b\in Q$. It is close to $x$ if $c<\frac{1}{100}$:

We have proved $P\cap B(0,10)\subset Q^{(r/2)}$ and thus $P^{(r/2)}\cap B(0,10)\subset Q^{(r)}$. Hence, $H$ is contained in the $(r,k)$-plate with central plane $Q$.

To prove the second property, we note that the set of $k$-planes in $\mathbb{A}(k,d)$ whose intersection with $B(0,10)$ is contained in a given $(s,k)$-plate is contained in an $O(s)$-ball $B$ of $\mathbb{A}(k,d)$. First suppose $B$ is contained within some coordinate chart; we would like to prove that $|\mathcal{N}\cap B|\lesssim\left(\frac{s}{r}\right)^{(k+1)(d-k)}$. To show this, note that $\{B(x,r):x\in\mathcal{N}\cap B\}$ is a packing of $B^{(r)}$ with finitely overlapping $r$-balls. Now map the chart to $\R^{(k+1)(d-k)}$. Since the map only distorts distances by a constant factor, we can pack $|\mathcal{N}\cap B|$ many finitely overlapping $c_{1}r$-balls into a ball of radius $O(s)$. Thus by a volume argument, we have $|\mathcal{N}\cap B|\lesssim\left(\frac{s}{r}\right)^{(k+1)(d-k)}$. Since there are finitely many charts, we can apply the argument to $B$ intersecting each chart, which proves the second property.

We specialize our discussion to tubes. For each scale $\delta$, let $\mathcal{T}^{\delta}$ be a cover of $[0,1]^{d}$ with $\delta$-tubes such that every $\frac{\delta}{2}$-tube (and in particular every $r$-tube with $r<\frac{\delta}{2}$) is contained in at least $1$ and at most $C_{d}$ many tubes of $\mathcal{T}^{\delta}$. Slightly abusing notation (á la [18]), we will also use $\mathcal{T},\mathcal{T}_{\delta},\mathcal{T}_{\Delta}$ to represent sets of tubes, where the subscript $\delta$ helpfully indicates a set of $\delta$-tubes.

In Theorem 1.13, we pay attention to certain plates with disproportionately much mass.

Other notation is following [18]. Unlike [18], we work with ordinary rather than dyadic tubes. The advantage of dyadic tubes is that every $2^{-n}$-tube is in a unique $2^{-m}$-tube if $n>m$; thus, dyadic tubes will avoid the $C_{d}$ loss incurred by the finitely overlapping cover $\mathcal{T}^{\delta}$. However, dyadic tubes have the disadvantage that they don’t behave well under rotations or dilations, and it would be more cumbersome to define $(\delta,s,C,k)$-sets of dyadic tubes (whereas the definition for ordinary tubes is more geometric). Thus, in principle it is possible to work with dyadic tubes and save on the $C_{d}$ loss, but it doesn’t affect our numerology in the end (since our losses will depend badly on $d$ anyway), so we chose to work with ordinary tubes throughout.

The following is a simpler interpretation of $(\delta,s,C,k)$-set if the tubes all pass through the same point.

We will need the following lemma from [18].

First, since $(\delta,\kappa,\delta^{-\varepsilon},k)$-sets are $(\delta,\kappa,\delta^{-\varepsilon},k^{\prime})$-sets for any $k^{\prime}<k$, we can assume that $k\leq s<k+1$. Next, since $(\delta,t,\delta^{-\varepsilon})$-sets are $(\delta,t^{\prime},\delta^{-\varepsilon})$-sets for $t^{\prime}<t$, we may assume $t\leq k+1$. In particular, we get $t-s\leq 1$, a useful assumption.

We record a useful geometric fact about $(r,k)$-plates.

We prove a classical estimate which can be viewed as Theorem 1.8 with $\varepsilon=0$. We won’t need the fact that $\mathcal{T}(p)$ is a $(\delta,\kappa,\delta^{-\varepsilon},k)$-set. The $d=2$ case is proven as Proposition 2.13 and Corollary 2.14 of [18]. For higher dimensions, the proof is similar and we sketch the details. Let $A\lessapprox_{\delta}B$ denote the inequality

The following corollary of Proposition 2.9 is the form we will use.