19. Roth's theorem II: Fourier analytic proof in the integers

TL;DR

Roth's Theorem states that the size of the largest 3AP-free subset of 1 through N is at most N divided by log log N. The proof involves using Fourier analysis and a density increment strategy.

Transcript

[SQUEAKING] [PAPER RUSTLING] [CLICKING] YUFEI ZHAO: Last time we started talking about Roth's theorem, and we showed a Fourier analytic proof of Roth's theorem in the finite field model. So Roth's theorem in F3 to the N. And I want to today show you how to modify that proof to work in integers. And this will be basically Roth's original proof of hi... Read More

Key Insights

• 🥶 Roth's Theorem provides a bound on the largest 3AP-free subset of 1 through N.
• ❓ The proof involves using Fourier analysis and a density increment strategy.

Questions & Answers

Q: What is Roth's Theorem?

Roth's Theorem states that the size of the largest 3AP-free subset of 1 through N is at most N divided by log log N.

Q: How is the proof of Roth's Theorem different for integers compared to the finite field model?

In the proof for integers, there are no subspaces, so the strategy involves finding a larger Fourier coefficient and restricting to a subprogression where density increment occurs.

Q: What are the key insights from the content?

1. Roth's Theorem provides a bound on the size of the largest 3AP-free subset of 1 through N.
2. The proof involves using Fourier analysis and a density increment strategy.
3. Higher-order Fourier analysis can be used to prove similar results for 4APs.

Summary & Key Takeaways

• Roth's Theorem states a bound on the size of the largest 3AP-free subset of 1 through N, which is at most N divided by log log N.

• The proof involves using Fourier analysis and a density increment strategy.

• The strategy includes finding a large Fourier coefficient and then finding a subprogression where there is a density increment.