---
title: '18.156 notes 03: projection theory for balls in euclidean space'
description: In this lecture, we develop the tools from the last lecture in the more geometric setting of Euclidean space.
date: '2025-02-11'
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---
In this lecture, we develop the tools from the last lecture in the more geometric setting of Euclidean space.

We first introduced our main tools in the setting of finite fields, where the technical details are simple. Now we adapt these tools to Euclidean space. Euclidean space has many different scales. We have to take into account many different scales in order to even ask good questions in Euclidean space. Paying attention to multiple scales will go on to be one of the key ideas in the subject.

We suppose that $X$ is a set of disjoint balls in Euclidean space, and study the orthogonal projections of $X$ in different directions. Here is the precise setup.

**SETUP**

Let $X$ be a set of disjoint unit balls in $B_R \subseteq \mathbb{R}^2$. Let $D \subset S^1$ be a set of $1/R$ separated directions.

$$
S=S(X,D)=\max_{\theta \in D}|\pi_\theta(X)|.
$$

$$
N_X(r)=\max_{c \in \mathbb{R}^2}|X \cap B(c,r)|.
$$

$$
N_D(\rho)=\max_{\substack{\sigma \in S' \text{arc}\\ |\sigma|=\rho}}|D \cap \sigma|.
$$

**Double Counting**

> \[!theorem|latex\] Theorem 3.1. Double Counting Real Version
>
> If **SETUP**, then
>
> $$
> |D|\lesssim \frac{S}{|X|}\sum_{1\leq r \leq R}N_X(r)N_D(1/r).
> $$
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec03_pdf/#page=1">

^theorem-3-1

![[courses/18-156-projection-theory-spring-2025/static_resources/latex/ballanglediagram.svg]]

_Figure 1._ Angle between two balls

^figure-1

> \[!proof|latex\]+ Proof of theorem 3.1.
>
> $$
> *=\#\{B_1,B_2 \text{ unit balls }\in X, \theta \in D: \pi_\theta(B_1) \cap \pi_\theta(B_2)\not=\emptyset\}.
> $$
>
> <u>Lower bound</u>: $*\gtrsim |D|\left(\frac{|X|}{S}\right)^2S$. It basically follows from the same argument as in the finite field setting.
>
> <u>Upper bound</u>: Fix $B_1,B_2$ with $dist(B_1,B_2)\sim r$, let $c(B_1)$, $c(B_2)$ be the center of $B_1$ and $B_2$. Write
>
> $$
> v=\frac{c(B_2)-c(B_1)}{|c(B_2)-c(B_1)|}
> $$
>
> to be the angle from $B_1$ to $B_2$. (see Figure [[courses/18-156-projection-theory-spring-2025/notes/03#^figure-1|1]]) If $\pi_\theta(B_1)\cap \pi_\theta(B_2)\not=\emptyset$, then $angle(\theta,\gamma)\lesssim 1/r$. Thus,
>
> $$
> \#\{\theta: \pi_\theta(B_1)\cap \pi_\theta(B_2)\}\lesssim N_D(1/r).
> $$
>
> $$
> \#\{B_1,B_2\in X: dist(B_1,B_2)\lesssim r\}\lesssim |X|N_X(r).
> $$
>
> Thus,
>
> $$
> * \lesssim \sum_{\substack{r \text{ dyadic }\\ 1 \leq r \leq R}} |X|N_X(r)N_D(1/r)
> $$
>
> so
>
> $$
> |X|^2S^{-1}D\lesssim * \lesssim \sum_{\substack{r \text{ dyadic }\\ 1 \leq r \leq R}}|X|N_X(r)N_D(1/r) .
> $$

^proof-3-1

> \[!example|latex\] Example 3.2
>
> For $N_X(r)$,
>
> 1. $X$ neighborhood of a curve. (see Figure [[courses/18-156-projection-theory-spring-2025/notes/03#^figure-2a|2a]])
>
>    $$
>    N_X(r) \sim r
>    $$
>
> 2. Well-spaced $N \times B$ grid. (see Figure [[courses/18-156-projection-theory-spring-2025/notes/03#^figure-2b|2b]])
>
>    $$
>    N_X(r)=\begin{cases}
>                1  & r \leq R/N \\
>                r^2\frac{N^2}{R^2} & r> R/N
>            \end{cases}
>    $$
>
> 3. A cluster of $N^2$ unit balls (see Figure [[courses/18-156-projection-theory-spring-2025/notes/03#^figure-2c|2c]])
>
>    $$
>    N_X(r)\sim \begin{cases}
>                r^2 & r \leq N\\
>                N^2 & r>N
>            \end{cases}
>    $$
>
> ![[courses/18-156-projection-theory-spring-2025/static_resources/latex/neighborhoodofcurvediagram.svg]]
>
> _Figure 2a._ A neighborhood of a curve.
>
> ^figure-2a
>
> ![[courses/18-156-projection-theory-spring-2025/static_resources/latex/wellspacedgriddiagram.svg]]
>
> _Figure 2b._ A well-spaced grid.
>
> ^figure-2b
>
> ![[courses/18-156-projection-theory-spring-2025/static_resources/latex/clusteredballsdiagram.svg]]
>
> _Figure 2c._ A cluster of balls.
>
> ^figure-2c
>
> _Figure 2._ Examples for $N_X(r)$.
>
> ^figure-2
>
> <u>Pictures of How $N_r(X)$ depends on $r$</u> (see Figure [[courses/18-156-projection-theory-spring-2025/notes/03#^figure-3|3]])
>
> ![[courses/18-156-projection-theory-spring-2025/static_resources/latex/nrxplotsdiagram.svg]]
>
> _Figure 3._ Plots of $N_r(X)$ vs $r$.
>
> ^figure-3
>
> Normalize $N=R^{1/2}$.
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec03_pdf/#page=2">

^example-3-2

<u>Straight Line Case </u>

$$
|X|=R^\alpha, N_X(r)\sim r^\alpha
$$

We call this regular $\alpha$ dim spacing.

<u>Below Straight Line Case </u>

$$
|X|=R^\alpha, N_X(r)\lesssim r^\alpha
$$

We call this $\alpha$ dim spacing.

> \[!definition|latex\] Definition 3.3.
>
> We say that $X$ has Hausdorff spacing if it has $\alpha$ dimension spacing for some $\alpha$. Another way to say this is that
>
> $$
> N_{R^\beta}(X)\lesssim |X|^\beta
> $$
>
> for any $0\leq \beta \leq 1$.
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec03_pdf/#page=3">

^definition-3-3

> \[!corollary|latex\] Corollary 3.4. Double Counting Real Version
>
> If **SETUP** $X$, $D$ has Hausdorff spacing then
>
> $$
> |D|\lesssim \log R (|S|+\frac{|S|}{|X|}|D|) \Rightarrow (S \sim X \text{ or }|D| \lesssim S).
> $$
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec03_pdf/#page=4">

^corollary-3-4

> \[!proof|latex\]+ Proof of corollary 3.4.
>
> Let’s calculate $N_X(r)N_D(1/r)$. Suppose $r=R^\beta$, the Hausdorff condition implies
>
> $$
> N_X(R^\beta)N_D(R^{-\beta})\lesssim |X|^\beta |D|^{1-\beta}
> $$
>
> Thus, by theorem [[courses/18-156-projection-theory-spring-2025/notes/03#^theorem-3-1|3.1]],
>
> $$
> |D| \lesssim \log R\frac{|S|}{|X|}\max_{0\leq \beta \leq 1}|X|^\beta |D|^{1-\beta}\lesssim \log R(|S|+\frac{|S|}{|X|})|D|.
> $$

^proof-3-4

Recall the theorem in the finite field case.

> \[!theorem|latex\] Theorem 3.5.
>
> If $X \subseteq \mathbb{F}_q^2$, $D \subseteq \mathbb{F}_q, S=\max_{\theta\in D}|\pi_\theta(X)|$ then $S \sim |X|$ or $|D|\lesssim S$.
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec03_pdf/#page=4">

^theorem-3-5

Note that in the $\mathbb{R}$ setting if we impose the Hausdorff spacing condition, then we get basically the same result as in the finite field case.

Now let’s compare result in projection theory in $\mathbb{F}_q^2$ vs unit balls in $B_R^2$ with Hausdorff spacing.

> \[!theorem|latex\] Theorem 3.6. Fourier Method Finite Field
>
> If $\mathbb{F}_q$-SETUP and $S \leq q/2$, then
>
> $$
> |D|\lesssim \frac{Sq}{|X|}.
> $$
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec03_pdf/#page=4">

^theorem-3-6

> \[!corollary|latex\] Corollary 3.7.
>
> If $\mathbb{R}$-SETUP and $X,D$ has Hausdorff spacing. Then, $|D|\lesssim \frac{SR}{|X|}$
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec03_pdf/#page=4">

^corollary-3-7

> \[!conjecture|latex\] Conjecture 3.8.
>
> If $p$ primes, $\mathbb{F}_p$ SETUP and $S \leq \frac{1}{2}\min(q,|X|)$ then
>
> $$
> |D| \lesssim \frac{|S|^2}{X}
> $$
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec03_pdf/#page=4">

^conjecture-3-8

> \[!conjecture|latex\] Conjecture 3.9. Furstenberg
>
> If SETUP, $X$ and $D$ has Hausdorff spacing and
>
> $$
> S \leq R^{-\epsilon}\min(R,|x|),
> $$
>
> then
>
> $$
> |D|\lessapprox \frac{|S|^2}{|R|}.
> $$
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec03_pdf/#page=4">

^conjecture-3-9

The above conjecture is proven in 2024 by (Orponen, Shmerkin, Ren and Wang)

## 3.1 fourier method

> \[!lemma|latex\] Lemma 3.10. Main lemma in finite field
>
> If $\mathbb{L}$ is a set of lines in $\mathbb{F}_q^2$. Write $f=\sum_{L \in \mathbb{L}}1_L(x)$. Then, $f=f_0+f_1$ so $\text{supp} \hat{f}_0=\{0\}$, $\text{supp} \hat{f}_h=\{0\}^c$. Then, $f_0$ is a constant function. Then, $\|f_0\|_2^2=|\mathbb{L}|^2$, $\|f_h\|_2^2=|\mathbb{L}|q$.
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec03_pdf/#page=5">

^lemma-3-10

Now, let’s look at the $\mathbb{R}$ setting. Let $\mathbb{T}$ be a set of $1 \times R$ in $\mathbb{R}^2$. Let $\phi_T$ be a smooth approximation of $1_T$.

> \[!lemma|latex\] Lemma 3.11. Main lemma in real
>
> Let $\mathbb{T}$ be a set of $1 \times R$ rectangles in $\mathbb{R}^2$. Let $f=\sum_{T \in \mathbb{T}} \phi_T(x)$. Then,
>
> $$
> f=\sum_{\substack{1\leq r \leq R\\ dyadic}}f_r(x)
> $$
>
> such that $\text{supp} \hat{f}_r \subseteq B(1/r)$ and $\|f_r\|_2^2\lesssim N_{\mathbb{T}}(r)|\mathbb{T}|r^{-1}R$ where
>
> $$
> N_\mathbb{T}(r):=\max_{\widetilde{T}: 2r \times 2R \text{rect}} \#\{T \in \mathbb{T}: T \subset \widetilde{T}\}.
> $$
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec03_pdf/#page=5">

^lemma-3-11

> \[!proof|latex\]+ Proof of lemma 3.11.
>
> (proof sketch of main lemma)
>
> $\text{supp} \hat{\phi}_T\subset T^*$ where $T^*:=\{\xi \in \mathbb{R}^2: |(x_1-x_2)\cdot \xi|\leq 1, \text{ any }x_1,x_2\in T\}$. (see Figure [[courses/18-156-projection-theory-spring-2025/notes/03#^figure-4|4]])
>
> ![[courses/18-156-projection-theory-spring-2025/static_resources/latex/dualrectanglediagram.svg]]
>
> _Figure 4._ The dual of a rectangle.
>
> ^figure-4

^proof-3-11

**Littlewood-Paley decomposition**

Write $1=\sum_{\substack{1\leq r \leq R\\ dyadic}} \eta_r(\xi)$ with $\eta_r \geq 0$ such that

$$
\text{supp} \eta_r \subseteq Ann(\frac{1}{10r} \leq |\xi| \leq \frac{1}{r}), 1 < r <R
$$

and $\text{supp} \eta_R \subseteq B(1/R)$ and $\text{supp} \eta_1 \subseteq \{\xi: |\xi|>1/10\}$. Define $f_r=(\eta_r \hat f)^\vee$ so $\text{supp} \hat{f_r} \subseteq B(1/r)$. In particular, we can write $\phi_{T,r}={(\eta_r \hat{\phi}_T)}^{\vee}$.

**Visual of $\hat{\eta_r}$ and $\phi_{T,r}$**

We have $\eta_r(\xi)\sim 1$ on $Ann(1/r)$ and

$$
|\check{\eta_r}(x)|\sim \begin{cases}
    1/r^2 \text{ on }|x|\lesssim r\\
    \text{rapidly decay if } |x|>r.
\end{cases}
$$

(see Figure [[courses/18-156-projection-theory-spring-2025/notes/03#^figure-5|5]])

![[courses/18-156-projection-theory-spring-2025/static_resources/latex/visualetardiagram.svg]]

_Figure 5._ Visual of radial component of $\hat{\eta_r}$.

^figure-5

where $\check{\eta_r}(x)=\int e^{i x \cdot \xi }\eta_r(\xi)d \xi$. Note that $\int \check{\eta_r}(x) dx =\eta_r(0)$. As $\int |\check{\eta_r}(x)|dx=\eta_r(0)$, we have that $f*|\check{\eta_r}(x)|\sim$ Average of $f$ on $B(x,r)$. As $\phi_{T,r}=\phi_T*\check{\eta_1}$, we have $|\phi_{T,r}(X)|\sim r^{-1} 1_{r \text{ neighborhood of } T}$. (see Figure [[courses/18-156-projection-theory-spring-2025/notes/03#^figure-6|6]])

![[courses/18-156-projection-theory-spring-2025/static_resources/latex/tubeneighborhooddiagram.svg]]

_Figure 6._ A tube $T$ and its $r$ neighborhood.

^figure-6

> \[!lemma|latex\] Lemma 3.12. Orthogonality
>
> If $T_1,T_2$ are $1 \times R$ tubes then $|\langle \phi_{T_1,r}, \phi_{T_2,r}\rangle|\lesssim R^{-1000}$ unless there exists $\widetilde{T}$, a $R^\epsilon r \times R^{1+\epsilon}$ rectangle such that $T_1, T_2 \in \widetilde{T}$.
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec03_pdf/#page=13">

^lemma-3-12

> \[!proof|latex\]+ Proof of lemma 3.12.
>
> (proof sketch) If $angle(T_1,T_2)\gtrsim R^\epsilon \frac{r}{R}$, then $\text{supp} \hat{\phi_{T_1,R}}\cap \text{supp} \hat{\phi_{T_2,R}}=\emptyset$. If $N_r(T_1)$ and $N_r(T_2)$ are disjoint, then
>
> $$
> \int \phi_{T_1,r}\phi_{T_2,r}=\int \phi_{T_1}*\hat{\eta_r}\phi_{T_1}*\hat{\eta_r} \lesssim R^{-1000}
> $$
>
> as $\phi_{T_1}*\hat{\eta_r}$ and $\phi_{T_1}*\hat{\eta_r}$ have essentially disjoint support.

^proof-3-12

$\mathbf{L^2}$ estimates

<a id="eq-10" class="latex-eq"></a><a id="eq-11" class="latex-eq"></a><a id="eq-12" class="latex-eq"></a><a id="eq-13" class="latex-eq"></a><a id="eq-14" class="latex-eq"></a><a id="eq-15" class="latex-eq"></a><a id="eq-16" class="latex-eq"></a>

$$
\begin{align*}
\|f_r\|_2^2&= \|\sum_{T \in \mathbb{T}}\phi_{T,r}\|_{L^2}^2 \tag{10} \\
&= \sum_{T_1,T_2} \langle \phi_{T_1,r}, \phi_{T_2,r}\rangle \tag{11} \\
&= \sum_{T_1 \sim_r T_2} \langle \phi_{T_1,r}, \phi_{T_2,r}\rangle + neglible \tag{12} \\
&\leq \sum_{T_1\sim_r T_2} \|\phi_{T,r}\|_2^2+\|\phi_{T_2,r}\|^2_{2} \tag{13} \\
&\leq N_r(\mathbb{T}) \sum_{T \in \mathbb{T}} \|\phi_{T,r}\|_2^2 \tag{14} \\
&= N_{\mathbb{T}}(r) \sum_{T \in \mathbb{T}}\|\phi_{T,r}\|_2^2 \tag{15} \\
&= N_{\mathbb{T}}(r)|\mathbb{T}|r^{-2}rR \tag{16}
\end{align*}
$$

where $r^{-2}$ is the amplitude and $rR$ is the area.