13.1 Higher-order Fourier analysis

Higher-order Fourier analysis takes classical Fourier analysis to the next level, studying functions on finite abelian groups. It's a game-changer in additive combinatorics, using Gowers uniformity norms to measure how close functions are to polynomials.

This powerful tool has cracked long-standing problems like Szemerédi's theorem and the Green-Tao theorem. It's also great for studying sumsets and additive configurations, showing how math can reveal hidden patterns in numbers.

Higher-order Fourier analysis

Introduction to higher-order Fourier analysis

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• Higher-order Fourier analysis generalizes classical Fourier analysis to study functions on finite abelian groups, particularly in additive combinatorics
• Focuses on analyzing the behavior of functions using Gowers uniformity norms, which measure how closely a function resembles a polynomial of a given degree
• Decomposes a function into a structured part (polynomial-like) and a pseudorandom part (uniform) for separate analysis
• Connects to various areas of mathematics, including ergodic theory, number theory, and theoretical computer science

Applications of higher-order Fourier analysis in additive combinatorics

• Successfully solves problems in additive combinatorics, such as finding arithmetic progressions in subsets of integers (Szemerédi's theorem)
• Studies the structure of sumsets and characterizes sets with small doubling property (Freiman-Ruzsa theorem)
• Establishes bounds on the size of subsets of finite abelian groups that avoid certain additive configurations (corners or simplices)
• Applies to the study of random sets and random matrices, leading to new results in probabilistic combinatorics and random matrix theory

Fourier coefficients and additive structures

Higher-order Fourier coefficients and Gowers uniformity norms

• Higher-order Fourier coefficients, or Gowers-Host-Kra (GHK) coefficients, capture the correlation between a function and polynomial phases
• Gowers uniformity norms, expressed in terms of higher-order Fourier coefficients, quantitatively measure a function's structure
• The distribution of higher-order Fourier coefficients reveals the presence of additive structures within a set (arithmetic progressions or polynomial patterns)
• The inverse theorem for Gowers uniformity norms states that a function with a large Gowers norm of a given degree must correlate with a polynomial phase of that degree, indicating an additive structure

Tools developed from studying higher-order Fourier coefficients

• Studying higher-order Fourier coefficients has led to powerful tools in additive combinatorics
• Density increment argument used to prove results related to the existence of additive structures in dense sets
• Energy increment method employed to analyze the structure of sets with small doubling property or to find patterns in subsets of finite abelian groups
• These tools have been instrumental in solving various problems in additive combinatorics and establishing important results (Szemerédi's theorem, Green-Tao theorem)

Applications in additive combinatorics

Proving Szemerédi's theorem and the Green-Tao theorem

• Higher-order Fourier analysis proves Szemerédi's theorem, which states that any subset of integers with positive upper density contains arbitrarily long arithmetic progressions
• The Green-Tao theorem, asserting the existence of arbitrarily long arithmetic progressions in the primes, is proved using higher-order Fourier analysis combined with analytic number theory techniques
• These results demonstrate the power of higher-order Fourier analysis in tackling long-standing problems in additive combinatorics and number theory

Studying the structure of sumsets and additive configurations

• Higher-order Fourier analysis is applied to study the structure of sumsets and prove the Freiman-Ruzsa theorem, which characterizes sets with small doubling property
• Techniques from higher-order Fourier analysis establish bounds on the size of subsets of finite abelian groups that avoid certain additive configurations (corners or simplices)
• These applications showcase the versatility of higher-order Fourier analysis in understanding the additive properties of sets and their subsets

Limitations and extensions of Fourier analysis

Limitations of higher-order Fourier analysis

• Higher-order Fourier analysis is most effective when dealing with linear or polynomial structures but has limited applicability to more general nonlinear patterns
• The bounds obtained through higher-order Fourier analysis are often not tight, and improving these bounds is an active research area
• Current methods of higher-order Fourier analysis are primarily effective for finite abelian groups, and extending these techniques to non-abelian groups or infinite-dimensional spaces remains challenging

Combining higher-order Fourier analysis with other tools

• Higher-order Fourier analysis has been combined with other tools from additive combinatorics to tackle more complex problems
• The polynomial method, which represents sets and functions using polynomials, can be used in conjunction with higher-order Fourier analysis to study more intricate additive structures
• The slice rank method, which measures the complexity of a tensor by its decomposition into simpler tensors, has been employed alongside higher-order Fourier analysis to solve problems in extremal combinatorics and number theory

Future directions and potential extensions

• Developing more efficient algorithms for computing Gowers uniformity norms is an ongoing research goal
• Exploring connections between higher-order Fourier analysis and other areas of mathematics (algebraic geometry, functional analysis) may lead to new insights and techniques
• Finding new applications of higher-order Fourier analysis in theoretical computer science and other fields is a promising avenue for future research
• Extending higher-order Fourier analysis to non-abelian groups or infinite-dimensional spaces could significantly expand its scope and applicability in solving a wider range of problems in additive combinatorics and beyond