7.3 Fourier analytic proof of Roth's theorem

Fourier analysis is a powerful tool in proving Roth's theorem on arithmetic progressions. It uses transforms and convolutions to study the indicator function of a set, revealing patterns that indicate the presence of arithmetic progressions.

The proof hinges on a density increment argument. By analyzing Fourier coefficients, we can either find many arithmetic progressions or increase the set's density on a subprogression, leading to an iterative process that eventually finds a progression.

Roth's Theorem on Arithmetic Progressions

Statement and Key Definitions

• Roth's theorem states every subset of the integers with positive upper density contains infinitely many arithmetic progressions of length 3
  • An arithmetic progression of length 3 is a set of integers {a, a+d, a+2d} where a represents the starting term and d represents the common difference
  • The upper density of a set A of integers is defined as $limsup_{N→∞} |A ∩ [1,N]| / N$, where $|A ∩ [1,N]|$ denotes the number of elements of A in the interval $[1,N]$

Significance and Connections

• Roth's theorem is a fundamental result in additive combinatorics
  • Has connections to other areas of mathematics such as number theory and ergodic theory
  • Provides a quantitative bound on the size of the largest subset of {1, 2, ..., N} without an arithmetic progression of length 3
  • Inspired further developments and techniques in the field (polynomial method, hypergraph removal lemma)

Fourier Analysis for Roth's Theorem

Fourier Transform and Convolution

• The proof of Roth's theorem uses Fourier analysis on the group $Z/NZ$, where N is a large integer
  • The Fourier transform of a function $f: Z/NZ → C$ is defined as $f^(r) = (1/N) ∑_{x∈Z/NZ} f(x) e^(-2πirx/N)$ for $r ∈ Z/NZ$
  • The convolution of two functions f and g on $Z/NZ$ is defined as $(f*g)(x) = ∑_{y∈Z/NZ} f(y) g(x-y)$

Indicator Function and L^2 Norm

• The key idea is to consider the indicator function $1_A$ of the set A and its Fourier transform $1^_A$
  • Study the L^2 norm $||1_A * 1_A * 1_A||_2$
  • If $||1_A * 1_A * 1_A||_2$ is large, then A contains many arithmetic progressions of length 3
  • If $||1_A * 1_A * 1_A||_2$ is small, then the Fourier transform $1^_A$ has a large L^∞ norm, which leads to a density increment on a subprogression

Density Increment Argument

Finding a Large Fourier Coefficient

• The density increment argument is a key step in the proof of Roth's theorem
  • If the L^2 norm $||1_A * 1_A * 1_A||_2$ is small, then there exists a Fourier coefficient $1^_A(r)$ with large magnitude
  • Using the large Fourier coefficient, one can find a long arithmetic progression P such that the density of A on P is significantly larger than the density of A in $Z/NZ$

Iterative Process

• The density increment allows for an iterative argument
  • If A does not contain an arithmetic progression of length 3, then we can find a subprogression where A has a higher density and repeat the argument
  • The iteration terminates when the density of A on a subprogression exceeds a certain threshold, at which point we can find an arithmetic progression of length 3 in A

Limitations of Fourier Analysis

Suboptimal Bounds

• The Fourier analytic proof of Roth's theorem provides a quantitative bound on the size of the largest subset of {1, 2, ..., N} without an arithmetic progression of length 3, but this bound is not optimal
  • The best-known upper bound on the size of such a subset is $O(N / (log N)^{1/4})$, while the Fourier analytic proof gives a weaker bound of $O(N / (log log N)^{1/2})$

Reliance on Large Fourier Coefficients

• The limitation of the Fourier analytic approach is that it relies on finding a large Fourier coefficient
  • Only guarantees a density increment on a long arithmetic progression
  • More advanced techniques have been used to improve the bounds on Roth's theorem (polynomial method, hypergraph removal lemma)