Combinatorial Number Theory

We focus on two proof techniques common in CNT: ergodic theoretic proofs and Fourier analytic proofs.
The main goal of the course is to introduce the students to the theory of higher order Fourier analysis.



Syllabus

1. Fundamentals of Ergodic theory:
• Ergodic proof of Fürstenberg-Sarkozy theorem for squares.
• Ergodic proof of Roth's theorem.
2. Fourier analysis over finite abelian groups:
• Fourier analytic proof of Fürstenberg-Sarkozy theorem.
• Fourier analytic proof of Roth's theorem.
3. Higher order Fourier analysis:
• Gowers norms.
• The Inverse U^3 theorem.
  
• The arithmetic regularity lemma of Green and Tao.
• The Gowers-Wolf decomposition.
  

List of lectures:
  
    Lecture 1 - Orbits of transformations.
    Lecture 2 - Measure preserving systems and ergodicity.
    Lecture 3 - Recurrence.
    Lecture 4 - The mean and pointwise ergodic theorems.
    Lecture 5 - Ergodicity and Mixing.
    Lecture 6 - Ergodic proof of Fürstenberg-Sarkozy theorem for squares.
    Lecture 7 - Ergodic proof of Roth's theorem.
    Lecture 8 - Fourier analytic proof of Roth's theorem.
    Lecture 9 - Fourier analytic proof of Fürstenberg-Sarkozy theorem for squares.
    Lecture 10 - Gowers norms
    Lecture 11 - The Inverse U^3 theorem - I
    Lecture 12 - The Inverse U^3 theorem - II
    Lecture 13 - The Inverse U^3 theorem - III
    Lecture 14 - The arithmetic regularity lemma for U^3 in F_p^n.