20. Roth's theorem III: polynomial method and arithmetic regularity

TL;DR

Roth's Theorem states that for every set A with a given density, there exists a common difference y such that the number of 3-AP's with y in A is close to what would be expected in a random set.

Transcript

YUFEI ZHAO: For the past couple lectures, we've been talking about Roth's theorem. And we showed-- so we saw a proof of Roth's theorem using Fourier analytic methods. And we saw basically the same proof but in two different settings. So two lectures ago, we saw a proof in F3 to the M And basically the same strategy, but with a bit more work, we wer... Read More

Key Insights

• 😫 Roth's Theorem provides a bound on the number of 3-AP's in a set with a certain density, using Fourier analytic and combinatorial methods.
• 🚾 The popular difference theorem strengthens Roth's Theorem by stating that there exists a popular common difference in 3-AP's, which is close to what would be expected in a random set.
• 🔄 The proof of the popular difference theorem involves an arithmetic regularity lemma and counting lemmas, leveraging the concept of pseudorandomness in functions.
• 🥹 The popular difference theorem holds for 3-AP's and 4-AP's but not for longer arithmetic progressions.

Questions & Answers

Q: What is Roth's Theorem?

Roth's Theorem states that for every set A with a given density, there exists a common difference y such that the number of 3-AP's with y in A is close to what would be expected in a random set.

Q: How is the proof of Roth's Theorem carried out?

The proof involves using Fourier analytic methods and the polynomial method in combinatorics to analyze the number of 3-AP's in a set A.

Q: What is the popular difference theorem?

The popular difference theorem states that for every epsilon, there exists a subspace U such that the number of 3-AP's with common difference restricted to U is close to what would be expected in a random set.

Q: How is the popular difference theorem proved?

The proof of the popular difference theorem involves an arithmetic regularity lemma and counting lemmas to analyze the 3-AP density and pseudorandomness of a function.

Summary & Key Takeaways

• Roth's Theorem provides a bound on the number of 3-AP's in a set A with a certain density.

• The proof of Roth's Theorem involves using Fourier analytic methods and the polynomial method in combinatorics.

• Another proof of Roth's Theorem is presented, which involves the use of popular differences in 3-AP's.

• The popular difference theorem states that for every epsilon, there exists a subspace U such that the number of 3-AP's with common difference restricted to U is close to what would be expected in a random set.

• The proof relies on an arithmetic regularity lemma and counting lemmas.