Gowers norms are powerful tools in additive combinatorics, measuring function 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 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 Uk(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 Uk(f) are defined inductively
U1(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∣≤∣g∣, then Uk(f)≤Uk(g)
Invariance under translation: Uk(f(x+t))=Uk(f(x)) for any t∈G
Invariance under multiplication by a character: Uk(f(x)χ(x))=Uk(f(x)) for any character χ
The U2 norm is related to the Fourier transform of the function
Higher norms capture more complex patterns and correlations
For example, the U3 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 Uk norm, then it must correlate with a structured object
Examples of structured objects include polynomial phase functions and
The inverse theorem for the U2 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 U3 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
: 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 Uk 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: