13.2 Inverse theorems for Gowers norms

Gowers norms are powerful tools in additive combinatorics, measuring function uniformity and pseudorandomness. They're key to studying arithmetic progressions and other patterns in sets. Understanding these norms helps uncover hidden structures in seemingly random data.

Inverse theorems for Gowers norms reveal the underlying structure of functions with large norms. These theorems connect to higher-order Fourier analysis and have solved long-standing problems in the field. They're essential for pushing the boundaries of additive combinatorics research.

Gowers Norms: Definition and Properties

Definition and Notation

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• Gowers norms, denoted as $U^k(G)$, are defined for functions on a finite Abelian group $G$ and a positive integer $k$
• Gowers norms measure the uniformity of a function
• The uniformity norms $U^k(f)$ are defined inductively
  • $U^1(f)$ is the absolute value of the average of $f$ over $G$
  • Higher norms are defined using the Gowers inner product

Properties of Gowers Norms

• Gowers norms satisfy several important properties
  • Monotonicity: if $|f| \leq |g|$, then $U^k(f) \leq U^k(g)$
  • Invariance under translation: $U^k(f(x+t)) = U^k(f(x))$ for any $t \in G$
  • Invariance under multiplication by a character: $U^k(f(x)\chi(x)) = U^k(f(x))$ for any character $\chi$
• The $U^2$ norm is related to the Fourier transform of the function
• Higher norms capture more complex patterns and correlations
  • For example, the $U^3$ norm is related to arithmetic progressions of length 3

Role in Additive Combinatorics

• Gowers norms play a crucial role in quantifying the uniformity and structure of functions in additive combinatorics
• They are used to study problems related to arithmetic progressions, sumsets, and other additive patterns
• Gowers norms provide a way to measure the "pseudorandomness" of a function or set
  • Functions with small Gowers norms are considered pseudorandom
  • Functions with large Gowers norms exhibit more structure and correlations

Inverse Theorems for Gowers Norms

Statement and Implications

• Inverse theorems for Gowers norms state that if a function has a large $U^k$ norm, then it must correlate with a structured object
  • Examples of structured objects include polynomial phase functions and nilsequences
• The inverse theorem for the $U^2$ norm is equivalent to the Fourier analytic proof of Roth's theorem on arithmetic progressions
• Higher-order inverse theorems, such as the inverse theorem for the $U^3$ norm, have important implications in additive combinatorics
  • They provide bounds on the density of sets avoiding certain patterns (arithmetic progressions of length 3)
• Inverse theorems for Gowers norms provide a powerful tool for understanding the structure of sets and functions in additive combinatorics

Proof Techniques

• The proofs of inverse theorems often involve deep techniques from various mathematical fields
  • Fourier analysis: studying the function's behavior in the frequency domain
  • Ergodic theory: analyzing the function's average behavior under translations
  • Number theory: exploiting the arithmetic properties of the underlying group
• The proofs typically involve decomposing the function into structured and pseudorandom components
• Bounds on the structured component are obtained using tools from the aforementioned fields
• The pseudorandom component is shown to have a small contribution to the Gowers norm

Gowers Norms vs Fourier Analysis

Higher-Order Fourier Analysis

• Higher-order Fourier analysis extends classical Fourier analysis to capture more complex patterns and correlations in functions
• It involves studying higher-order Fourier coefficients and nilsequences
• Higher-order Fourier analysis has led to significant progress in understanding the structure of sets and functions in additive combinatorics

Connection to Gowers Norms

• Gowers norms are closely related to the concepts of higher-order Fourier coefficients and nilsequences
• The $U^k$ norm of a function can be expressed in terms of its higher-order Fourier coefficients
  • This provides a link between the two concepts
• The connection between Gowers norms and higher-order Fourier analysis has opened up new avenues for research
  • It has led to the resolution of several long-standing problems in additive combinatorics (such as the cap set problem)

Applications of Inverse Theorems in Additive Combinatorics

Density Bounds for Additive Patterns

• Inverse theorems for Gowers norms can be used to establish density bounds for sets avoiding certain additive patterns
  • Examples of additive patterns include arithmetic progressions and more complex configurations
• The strategy typically involves assuming the set has a density above a certain threshold
  • Then, using the inverse theorem to show that the set must contain the desired pattern
• In some cases, the application of inverse theorems may require decomposing the set or function into structured and pseudorandom components

Combination with Other Techniques

• Inverse theorems can be combined with other techniques to obtain stronger results
  • Density increment arguments: iteratively finding subsets with increased density
  • Energy increment arguments: iteratively finding subsets with increased Fourier coefficients
• Applying inverse theorems for Gowers norms often requires careful analysis and estimates of various parameters
  • It also requires an understanding of the underlying additive structure of the problem at hand
• Inverse theorems have been used to solve problems related to:
  • Szemerédi's theorem on arithmetic progressions
  • The density Hales-Jewett theorem
  • The cap set problem