---
title: '18.156 notes 11: contagious structure in projection theory'
description: This technique will play an important role in the proof of the Bourgain-Katz-Tao projection theorem.
date: '2025-03-20'
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created: '2025-03-20'
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Suppose that $X \subset \mathbb{F}_q$. Recall that the exceptional directions are directions $\theta$ so that $\pi_\theta(X)$ is very small. In this class we explore the algebraic structure of the set of exceptional directions. A basic example is that $X$ is a square grid. In this case, the exceptional directions are rational numbers with small numerator / denominator. (The smaller the height of the rational number, the smaller $|\pi_\theta(X)|$ is. Notice that this set of exceptional directions has a lot of algebraic structure: the sum or product of two exceptional directions is also (pretty) exceptional. We call this contagious structure. Using combinatorial number theory, we show that for any set $X$, the set of exceptional directions has contagious structure. This idea builds on work of Edgar-Miller and was developed by Bourgain-Katz-Tao.

This technique will play an important role in the proof of the Bourgain-Katz-Tao projection theorem.

## 11.1 contagious structure lemma

> \[!lemma|latex\] Lemma 11.1.
>
> If $Z$ is an abelian group and $A \subset Z$, and
>
> $$
> |A-tA|\leq K|A| \text{ and }|A-t_2A|\leq K|A|,
> $$
>
> then $|A-(t_1\cdot t_2 )A|\leq K^2A$.
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec11_pdf/#page=1">

^lemma-11-1

> \[!proof|latex\]+ Proof of lemma 11.1.
>
> Note that $|A-t_2A|\leq K|A|$ implies that $|t_1A-t_1t_2A|\leq K|A|$. Let $\bar{B}=A$, $\bar{C}=t_1t_2A$, $\bar{A}=A$ in Rusza’s inequality, so
>
> $$
> |t_1A||A-t_1t_2A|\leq |t_1A-A||t_1-t_1t_2A|.
> $$
>
> Thus, $|A||A-t_1t_2A|\leq K^2|A|$.

^proof-11-1

> \[!lemma|latex\] Lemma 11.2.
>
> If $|A+tA|\leq K|A|$ then $|A-tA| \leq K^2|A|$.
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec11_pdf/#page=1">

^lemma-11-2

> \[!proof|latex\]+ Proof of lemma 11.2.
>
> By Rusza’s inequality, $|A||A-tA|\leq |A+A||A+tA|$. By Plunnecke’s inequality, $|A+A|\leq K^2|A|$. Thus, $|A||A-tA|\leq K^3|A|$.

^proof-11-2

> \[!lemma|latex\] Lemma 11.3.
>
> If $|A+t_1A| \leq |A|$ and $|A+t_2A| \leq K|A|$, then
>
> $$
> |A+(t_1+t_2)A|\leq K^5|A|.
> $$
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec11_pdf/#page=1">

^lemma-11-3

> \[!proof|latex\]+ Proof of lemma 11.3.
>
> Note that $|A+(t_1+t_2)A| \leq |A+t_1A+t_2A|$. By the main lemma, there exists $X_1\subset A$ so for any $C$, we have $|X_1+C+t_1A|\leq |X_1+C|$ and there exists $X_2\subset A$ so for any $C$, we have $|X_2+C+t_1A|\leq |X_2+C|$.
>
> $$
> \begin{align*}
> |A+t_1A+t_2A| &\leq |x_1+x_2+A+t_1A+t_2A| \\
> &\leq K|x_1+x_2+A+t_2A| \\
> &\leq K^2|x_1+x_2+A| \\
> &\leq K^2|A+A+A| \\
> &\leq K^5|A|
> \end{align*}
> $$

^proof-11-3

Our goal today is to prove the following theorem.

> \[!theorem|latex\] Theorem 11.4.
>
> If $A\subset \mathbb{F}_p$, $|A|=p^{s_A}$, $D\subset \mathbb{F}_p$, $|D|=p^{s_D}$, $0 <s_A,s_p<1$. Then, there exists $\epsilon(s_A,s_D)>0$, $\max(s_A,s_D)>0$, $\max(|A+tA|)\geq p^{s_A+\epsilon(s_A,s_D)}$.
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec11_pdf/#page=2">

^theorem-11-4

> \[!corollary|latex\] Corollary 11.5.
>
> $|A+A\cdot A|\geq p^{s_A+\epsilon}$.
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec11_pdf/#page=2">

^corollary-11-5

Now, let’s recall double counting result.

> \[!lemma|latex\] Lemma 11.6. Double Counting
>
> Suppose $X \subseteq \mathbb{F}_p^2$, and $D \subseteq \mathbb{F}_p$, then
>
> $$
> \max_{t \in D}|\pi_t(X)| \gtrsim \min(|X|,|D|).
> $$
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec11_pdf/#page=2">

^lemma-11-6

Note that if $s_D >s_A$, then double counting implies theorem [[courses/18-156-projection-theory-spring-2025/notes/11#^theorem-11-4|11.4]], so the hard cases are the cases in which $0<s_D<s_A$. Let’s also recall a corollary from the previous section.

> \[!lemma|latex\] Lemma 11.7.
>
> If $0<s<t<1$, then there exists $k=k(s,t)$ so if $A \subseteq \mathbb{F}_p$, $|A|=p^s$ then $|poly_k(A)|\geq p^t$.
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec11_pdf/#page=2">

^lemma-11-7

The proof idea is to use lemma [[courses/18-156-projection-theory-spring-2025/notes/11#^lemma-11-7|11.7]] to increase $s_D$ to be bigger than $s_A$ by taking sums and products and then use the contagious structure.

> \[!proof|latex\]+ Proof of lemma 11.7.
>
> By lemma [[courses/18-156-projection-theory-spring-2025/notes/11#^lemma-11-7|11.7]], there exists $K(s_A,s_D)$ so $|poly_k(D)|>p^{s_A+r}$. By double counting there exists $u \in poly_k(D)$ so $|A+uA|>p^{s_A+r}$. But if $\max_{t\in D}|A+tA|\leq K|A|$, then the contagious structure says that
>
> $$
> \max_{u \in poly_k(D)}|A+uA|\leq K^{c(k)}|A|=K^{c(s_A,s_D)}|A|=K^cp^s.
> $$
>
> However, this would imply that $K^c\geq p^r$ which would imply that $p^{r/c}=p^\epsilon$ a contradiction.

^proof-11-7

The above theorem [[courses/18-156-projection-theory-spring-2025/notes/11#^theorem-11-4|11.4]] is a special case of the following theorem when we put $X=A\times A$.

> \[!theorem|latex\] Theorem 11.8. BKT
>
> If $X \subseteq \mathbb{F}_p^2$, $|X|=p^{s_X}$ with $0<s_X<2$, and $D \subseteq \mathbb{F}_p$ with $|D|=p^{s_D}$ such that $0<s_D$.
>
> $$
> \max_{t\in D}|\pi_t(X)|\geq p^\epsilon |X|
> $$
>
> <ref slug="courses/18-156-projection-theory-spring-2025/resources/mit18_156_s25_lec11_pdf/#page=3">

^theorem-11-8