This was my final paper for 18.099, seminar in discrete analysis, jointly with Sammy Luo and Ryan Alweiss.

We prove that every sufficiently large odd integer can be written as the sum of three primes, conditioned on a strong form of the prime number theorem.

1. Introduction

In this paper, we prove the following result:

Theorem 1 (Vinogradov)

Every sufficiently large odd integer $N$ is the sum of three prime numbers.

In fact, the following result is also true, called the “weak Goldbach conjecture”.

Theorem 2 (Weak Goldbach conjecture)

Every odd integer $N \ge 7$ is the sum of three prime numbers.

The proof of Vinogradov’s theorem becomes significantly simpler if one assumes the generalized Riemann hypothesis; this allows one to use a strong form of the prime number theorem (Theorem 9). This conditional proof was given by Hardy and Littlewood in …

EDIT: Here’s a July 19 draft that fixes some of the glaring issues that were pointed out.

This morning I finally uploaded the first drafts of my Napkin project, which I’ve been working on since December 2014. See the Napkin tab above for a listing of all drafts.

Napkin is my personal exposition project, which unifies together a lot of my blog posts and even more that I haven’t written on yet into a single coherent narrative. It’s written for students who don’t know much higher math, but are curious and already are comfortable with proofs. It’s especially suited for e.g. students who did contests like USAMO and IMO.

There are still a lot of rough edges in the draft, but I haven’t been able to find much time to work on it this whole calendar year, and so I’ve finally …

In this post I’ll describe the structure theorem over PID’s which generalizes the following results:

• Finite dimensional vector fields over $k$ are all of the form $k^{\oplus n}$,
• The classification theorem for finitely generated abelian groups,
• The Frobenius normal form of a matrix,
• The Jordan decomposition of a matrix.

1. Some ring theory prerequisites

(Prototypical example for this section: $R = \mathbb Z$.)

Before I can state the main theorem, I need to define a few terms for UFD’s, which behave much like $\mathbb Z$: Our intuition from the case $R = \mathbb Z$ basically carries over verbatim. We don’t even need to deal with prime ideals and can factor elements instead.

Definition 1. If $R$ is a UFD, then $p \in R$ is a …

This is a transcript of a talk I gave as part of MIT’s 18.434 class, the “Seminar in Theoretical Computer Science” as part of MIT’s communication requirement. (Insert snarky comment about MIT’s CI-* requirements here.) It probably would have made a nice math circle talk for high schoolers but I felt somewhat awkward having to present it to a bunch of students who were clearly older than me.

1. Preliminaries

1.1. Modular arithmetic

In middle school you might have encountered questions such as

Exercise 1. What is $3^{2016} \pmod{10}$?

You could answer such questions by listing out $3^n$ for small $n$ and then finding a pattern, in this case of period $4$. However, for large moduli this “brute-force” approach can be time-consuming.

Fortunately, it …

For some reason several classes at MIT this year involve Fourier analysis. I was always confused about this as a high schooler, because no one ever gave me the “orthonormal basis” explanation, so here goes. As a bonus, I also prove a form of Arrow’s Impossibility Theorem using binary Fourier analysis, and then talk about the fancier generalizations using Pontryagin duality and the Peter-Weyl theorem.

In what follows, we let $\mathbb T = \mathbb R/\mathbb Z$ denote the “circle group”, thought of as the additive group of “real numbers modulo $1$”. There is a canonical map $e : \mathbb T \rightarrow \mathbb C$ sending $\mathbb T$ to the complex unit circle, given by $e(\theta) = \exp(2\pi i \theta)$

I will tell you a story about the Reciprocity Law. After my thesis, I had the idea to define $L$-series for non-abelian extensions. But for them to agree with the $L$-series for abelian extensions, a certain isomorphism had to be true. I could show it implied all the standard reciprocity laws. So I called it the General Reciprocity Law and tried to prove it but couldn’t, even after many tries. Then I showed it to the other number theorists, but they all laughed at it, and I remember Hasse in particular telling me it couldn’t possibly be true.

Still, I kept at it, but nothing I tried worked. Not a week went by — for three years! — that I did not try to prove the Reciprocity Law. It was discouraging, and meanwhile I turned to other things. Then one afternoon I had nothing …

One of the pieces of advice I constantly give to young students preparing for math contests is that they should probably do harder problems. But perhaps I don’t preach this zealously enough for them to listen, so here’s a concrete reason (with actual math!) why I give this advice.

1. The AIME and USAMO

In the USA many students who seriously prepare for math contests eventually qualify for an exam called the AIME (American Invitational Math Exam). This is a 3-hour exam with 15 short-answer problems; the median score is maybe about 5 problems.

Correctly solving maybe 10 of the problems qualifies for the much more difficult USAMO. This national olympiad is much more daunting, with six proof-based problems given over nine hours. It is not uncommon for olympiad contestants to not solve a single problem (this certainly happened to me a fair share of times!).

You’ll …

As part of the 18.099 Discrete Analysis reading group at MIT, I presented section 4.7 of Tao-Vu’s Additive Combinatorics textbook. Here were the notes I used for the second half of my presentation.

1. Synopsis

We aim to prove the following result.

Theorem 1 (Bourgain)

Assume $N \ge 2$ is prime and $A, B \subseteq Z = \mathbb Z_N$. Assume that $$\delta \gg \sqrt{\frac{(\log \log N)^3}{\log N}}$$ is such that $\min\left\{ \mathbf P_ZA, \mathbf P_ZB \right\} \ge \delta$. Then $A+B$ contains a proper arithmetic progression of length at least $$\exp (\mathrm{C\; \dots }$$

Some time ago I was reading the 18.785 analytic NT notes to try and figure out some sections of Davenport that I couldn’t understand. These notes looked nice enough that I decided I should probably print them out, But much to my annoyance, I found that almost all the top margins were too tiny, and the bottom margins too big. (I say “almost all” since the lectures 19 and 24 (Bombieri proof and elliptic curves) were totally fine, for inexplicable reasons).

Thus, instead of reading Davenport like I told myself to, I ended up learning enough GhostScript flags to write the following short script, which I’m going to share today so that other people can find better things to do with their time.

#!/bin/bash
for file in $@
  do
    echo "Shifting $file ..."
    gs \
      -sDEVICE=pdfwrite \
      -o shifted-$file \
 -dPDFSETTINGS=/prepress \
 -c "<</PageOffset [0 -36]>> setpagedevice" \
 -f $file …

As part of the 18.099 discrete analysis reading group at MIT, I presented section 4.7 of Tao-Vu’s Additive Combinatorics textbook. Here were the notes I used for the first part of my presentation.

1. Synopsis

In the previous few lectures we’ve worked hard at developing the notion of characters, Bohr sets, spectrums. Today we put this all together to prove some Szemerédi-style results on arithmetic progressions of $\mathbb Z_N$.

Recall that Szemerédi’s Theorem states that:

Theorem 1 (Szemerédi)

Let $k \ge 3$ be an integer. Then for sufficiently large $N$, any subset of $\{1, \dots, N\}$ with density at least $$\frac{1}{(\log \log N)^{2^{-2^k+9}}}$$ contains a length $\mathrm{k\; \dots }$