--- title: 18.156 problem set 01 description: Please email your work by the morning of Thursday Feb 15 to me. tags: - course - mit - ocw - problem-set cssclasses: - course-notes license: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International license_url: https://creativecommons.org/licenses/by-nc-sa/4.0/ latex: source: static_resources/mit18_156_s25_pset1.tex lines: - 24 - 156 pset: 1 pdf: mit18_156_s25_pset1_pdf pageLayout: default slug: courses/18-156-projection-theory-spring-2025/notes/pset-01 permalink: https://aarnphm.xyz/courses/18-156-projection-theory-spring-2025/notes/pset-01.md generator: quartz: v4.6.0 hostedProvider: Cloudflare baseUrl: aarnphm.xyz full: https://aarnphm.xyz/llms-full.txt --- Please email your work by the morning of Thursday Feb 15 to me. **Letter of introduction.** So I can start getting to know you, could you send me a letter introducing yourself. - What math directions are you studying? - Tell me about your background in real analysis. What was your last real analysis class like? What parts did you find interesting? What parts did you find confusing or frustrating? - Are there any particular topics or themes in the course description that would be interesting or helpful for you? - How is the pace of the class for you so far? - Anything else on your mind about the class? ## digesting material from class Notation. Recall that we write $A \lesssim B$ to mean that there exists a constant $c$ so that $A \le c B$. **problem 1.** (Practicing Cauchy-Schwarz and double counting) Suppose that $X, Y$ are finite sets and $f: X \rightarrow Y$. If $|Y| \le (1/2) |X|$ prove that ^problem-1 $$ \# \{ x_1, x_2 \in X: f(x_1) = f(x_2) \} \gtrsim |X|^2 |Y|^{-1}. $$ We recall the convenitions and basic theorems about the Fourier transform over finite fields. Let $\mathbb{F}_q$ be the finite field with $q$ elements, where $q = p^r$ and $p$ prime. For $x, \xi \in \mathbb{F}_q^d$, we define $x \cdot \xi = x_1 \xi_1 + ... + x_d \xi_d$. We let $e: \mathbb{F}_q \rightarrow \mathbb{C}^*$ be a non-trivial group homomorphism, from $\mathbb{F}_q$ with addition to $\mathbb{C}^*$ with multiplication. (If $q =p$ is prime, then we can define $e(x) = e^{2 \pi i \frac{x}{p}}$. But any choice of $e$ works equally well. If $q = p^r$, the image of $e$ will be the $p^{th}$ roots of unity.) Suppose $f: \mathbb{F}_q^d \rightarrow \mathbb{C}$. For any $\xi \in \mathbb{F}_q^d$, define $$ \hat f(\xi) = \sum_{x \in \mathbb{F}_q^d} f(x) e( - x \cdot \xi). $$ > \[!theorem|latex\] Theorem 1. Fourier inversion > > If $f: \mathbb{F}_q^d \rightarrow \mathbb{C}$, then > > $$ > f(x) = \frac{1}{q^d} \sum_{\xi \in \mathbb{F}_q^d} \hat f(\xi) e(x \cdot \xi). > $$ ^theorem-1 > \[!theorem|latex\] Theorem 2. Plancherel > > If $f, g: \mathbb{F}_q^d \rightarrow \mathbb{C}$, then > > $$ > \sum_{x \in \mathbb{F}_q^d} f(x) \overline{g(x)} = \frac{1}{q^d} \sum_{\xi \in \mathbb{F}_q^d} \hat f(\xi) \overline{\hat g(\xi)}. > $$ ^theorem-2 **problem 2.** On your own, work through the proof of Theorem 1 and Theorem 2. (You don’t have to turn anything in.) The proofs are based on linear algebra and on orthogonality. Here is a sketchy outline. ^problem-2 a. Because $e: \mathbb{F}_q \rightarrow \mathbb{C}^*$ is a non-trivial group homomorphism, $\sum_{x \in \mathbb{F}_q} e(x) = 0$. b. Building on part a, show that for any non-zero $\xi$ in $\mathbb{F}_q^d$, $\sum_{x \in \mathbb{F}_q^d} e(x \cdot \xi) = 0$. c. Building on part b, show that for any $\xi_1, \xi_2 \in \mathbb{F}_q^d$, $$ \frac{1}{q^d} \sum_{x \in \mathbb{F}_q^d} e(x \cdot \xi_1) \overline{ e(x \cdot \xi_2)} = \begin{cases} 1 & \xi_1 = \xi_2 \\ 0 & \text{ else } \end{cases} $$ d. Show that the set of functions $\{ \frac{1}{q^{d/2}} e(x \cdot \xi) \}_{\xi \in \mathbb{F}_q^d}$ forms an orthonormal basis of $\ell^2(\mathbb{F}_q^d)$. e. Theorems 1 and 2 follow from d with a bit of algebra. **problem 3.** Suppose that $P \subset \mathbb{F}_q^d$ is an affine $k$-plane. Define the perpendicular subspace $P^\perp$ by ^problem-3 $$ P^\perp = \{ \xi \in \mathbb{F}_q^d: (x_1 - x_2) \cdot \xi = 0 \forall x_1, x_2 \in P \}. $$ Show that $$ | \hat 1_P(\xi)| = \begin{cases} q^k & \xi \in P^\perp \\ 0 & \text{ else} \\ \end{cases} $$ In class we proved some projection estimates for sets $X \subset \mathbb{F}_q^2$ using Fourier analysis. In this problem, you will generalize these estimates to $X \subset \mathbb{F}_q^3$ Let $Gr_q(k, d)$ be the set of $k$-dimensional subspaces $W \subset \mathbb{F}_q^d$. For each $W \in Gr_q(d-k,d)$, let $\pi^W: \mathbb{F}_q^d \rightarrow \mathbb{F}_q^k$ be a linear map with kernel $W$. **problem 4.** Suppose that $X \subset \mathbb{F}_q^3$. Suppose that $D \subset Gr_q(1,3)$. Let $S = S(X,D) = \max_{W \in D} | \pi^W(X)|$. ^problem-4 If $S \le q^2/2$, prove that $$ |D| \lesssim \frac{S q^2}{|X|}. \tag{1} $$ Then on your own, check that this gives sharp bounds in the following two cases. Case 1. If $D = Gr_q(1,3)$, then we get $|X| \lesssim S$, which is sharp. Case 2. If $X$ is a 2-plane in $\mathbb{F}_q^3$ and $D$ is the set of directions tangent to the 2-plane. ## optional exploring futher Here are some options for exploring further if you would like to do so. In each problem set, I will include some options to explore. At the end of the class, everyone will do a final project. One option for a final project is to study one of these further exploration questions and write up what you learn about it. Suppose that $X \subset \mathbb{F}_p^3$ and $D$ and $S$ are as above. Given $|X|$ and $S$, how big could $|D|$ be? Try out a few examples and see if you can make a plausible conjecture. If you want, you could also consider the same problem over $\mathbb{F}_q$, where there are more examples based on subfields. The example where $X$ is a 2-plane plays an important role in the last story. You could consider sets $X \subset \mathbb{F}_q^3$ where $|X \cap P| \le B$ for any 2-plane $P \subset \mathbb{F}_q^3$. Given $|X|$ and $S$ and $B$, how big could $|D|$ be? Again you could try some examples. Can you prove a bound that improves on problem 4 when $B$ is much less than $q^2$?