Glossary

13.2 Inverse theorems for Gowers norms

4 min read•july 30, 2024

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)U^k(G), are defined for functions on a finite Abelian group GG and a positive integer kk
  • Gowers norms measure the uniformity of a function
  • The Uk(f)U^k(f) are defined inductively
    • U1(f)U^1(f) is the absolute value of the average of ff over GG
    • Higher norms are defined using the Gowers inner product

Properties of Gowers Norms

  • Gowers norms satisfy several important properties
    • Monotonicity: if ∣f∣≤∣g∣|f| \leq |g|, then Uk(f)≤Uk(g)U^k(f) \leq U^k(g)
    • Invariance under translation: Uk(f(x+t))=Uk(f(x))U^k(f(x+t)) = U^k(f(x)) for any t∈Gt \in G
    • Invariance under multiplication by a character: Uk(f(x)χ(x))=Uk(f(x))U^k(f(x)\chi(x)) = U^k(f(x)) for any character χ\chi
  • The U2U^2 norm is related to the Fourier transform of the function
  • Higher norms capture more complex patterns and correlations
    • For example, the U3U^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 UkU^k norm, then it must correlate with a structured object
    • Examples of structured objects include polynomial phase functions and
  • The inverse theorem for the U2U^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 U3U^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
    • : 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 UkU^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:
    • on arithmetic progressions
    • The density Hales-Jewett theorem
    • The cap set problem

Key Terms to Review (17)

Additive Number Theory: Additive number theory is a branch of mathematics that focuses on the properties and behaviors of sets of integers under addition. This field investigates how various sets can be added together, often exploring the structure, distribution, and sums of these sets, and connects deeply with combinatorial techniques and problems in number theory.
Arithmetic Progression: An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This concept plays a crucial role in various areas of mathematics, including additive combinatorics, as it helps in understanding the structure and distribution of numbers within sets.
Ben Green: Ben Green is a prominent mathematician known for his groundbreaking contributions to additive combinatorics, particularly in relation to prime numbers and arithmetic progressions. His work has significantly influenced various areas of mathematics, including the development of new methods that intersect with concepts like uniformity norms and inverse theorems, revealing deeper connections between different mathematical frameworks.
Bourgain's Result: Bourgain's Result refers to significant findings in additive combinatorics that connect the behavior of functions with specific structural properties to their average behavior. This result emphasizes how certain uniformities in a function imply constraints on its complexity, revealing deep relationships between various areas such as harmonic analysis and number theory.
Concentration of Measure: Concentration of measure is a phenomenon in probability theory and statistics where a function of many independent random variables is likely to be close to its expected value, especially as the dimension increases. This idea plays a significant role in understanding the behavior of functions in high-dimensional spaces, which is crucial when dealing with Gowers norms and their inverse theorems.
Energy Method: The energy method is a powerful analytical tool used in additive combinatorics to study the structure of functions and sets, particularly in relation to Gowers norms. It involves examining the 'energy' of a configuration or function, which measures how much of a certain structure is present in the data. This technique helps in proving inverse theorems by connecting the behavior of functions under Gowers norms to their structural properties.
Fourier Analysis: Fourier analysis is a mathematical technique used to decompose functions into their constituent frequencies, facilitating the study of periodic functions and their properties. This method is essential in understanding various concepts in additive combinatorics, as it provides tools to analyze the structure and behavior of functions over groups, especially in the context of sums and products.
Gowers u^s norm: The Gowers u^s norm is a specific mathematical tool used to measure the uniformity of a function over a finite abelian group, capturing the concept of how 'random' or 'structured' the function is. This norm is crucial in additive combinatorics and helps to identify patterns within functions by analyzing their higher-order Fourier coefficients, thus providing insights into additive properties of sets and sequences.
Gowers' Inverse Theorem: Gowers' Inverse Theorem is a fundamental result in additive combinatorics that provides a characterization of functions with high Gowers norms. Essentially, it states that if a function has a sufficiently large Gowers norm, then it must exhibit a certain level of structured behavior, such as being close to a polynomial phase. This theorem links the concept of uniformity in functions to the existence of arithmetic progressions and other combinatorial structures.
Higher-order Fourier analysis: Higher-order Fourier analysis extends the classical Fourier analysis framework to study functions and sequences by examining their behavior under higher-order notions of uniformity and structure. This field focuses on analyzing additive combinatorial structures through Gowers norms, revealing deeper insights about the presence of patterns and regularities in sequences.
Nilsequences: Nilsequences are a special type of sequence that arise in additive combinatorics, particularly in the study of patterns within sequences of integers. They can be thought of as a tool to analyze the structure of sequences that exhibit certain regularities, often tied to the Gowers norms. These sequences help in understanding how additive properties behave under various conditions, especially when dealing with inverse theorems for Gowers norms.
Randomness extraction: Randomness extraction is a process that transforms a source of weak randomness into a source of nearly uniform random bits. This technique is vital in areas such as cryptography, algorithm design, and data compression, as it enables the construction of robust random sources from imperfect or biased inputs. By ensuring that the output is as close to true randomness as possible, randomness extraction plays a crucial role in applications involving expanders and extractors, as well as in understanding various inverse theorems related to Gowers norms.
Sum-free set: A sum-free set is a subset of integers such that no two elements in the set can be added together to form another element in the same set. This concept connects to various areas in combinatorics, particularly in understanding the structure and properties of sets of integers. Sum-free sets have significant implications in additive number theory and can be related to the study of higher-order additive properties, such as those explored in Gowers norms and more complex combinatorial configurations.
Szemerédi's theorem: Szemerédi's theorem states that for any positive integer $k$, any subset of the integers with positive density contains a non-empty subset of $k$ elements that form an arithmetic progression. This theorem is significant as it connects combinatorial number theory with additive combinatorics and has wide implications in various mathematical fields.
Terence Tao: Terence Tao is a renowned Australian-American mathematician known for his contributions to various areas of mathematics, including additive combinatorics. His work has significantly advanced the field, particularly in understanding prime numbers and combinatorial structures through innovative techniques and deep insights.
Uniformity: Uniformity refers to the concept of a certain level of regularity or consistency within a mathematical structure, particularly in the context of functions and sequences. In additive combinatorics, uniformity measures how evenly distributed certain algebraic or combinatorial properties are across a set, which plays a crucial role in understanding patterns and behaviors within number systems and sequences.
Uniformity Norms: Uniformity norms are a set of mathematical tools used in additive combinatorics to measure the uniformity of functions, particularly in relation to their behavior under certain transformations. They help in understanding the structure of sets through their additive properties by quantifying how far a function deviates from being uniformly distributed. This concept is key for studying additive structures and has significant applications in various proofs and theorems related to set addition and combinatorial number theory.
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