Foundations and Trends® in Theoretical Computer Science > Vol 13 > Issue 4

Higher-order Fourier Analysis and Applications

Hamed Hatami, McGill University, Canada, hatami@cs.mcgill.ca Pooya Hatami, Ohio State University, USA, pooyahat@gmail.com Shachar Lovett, UC San Diego, USA, slovett@cs.ucsd.edu
 
Suggested Citation
Hamed Hatami, Pooya Hatami and Shachar Lovett (2019), "Higher-order Fourier Analysis and Applications", Foundations and Trends® in Theoretical Computer Science: Vol. 13: No. 4, pp 247-448. http://dx.doi.org/10.1561/0400000064

Published: 26 Sep 2019
© 2019 H. Hatami, P. Hatami and S. Lovett
 
Subjects
Computational complexity,  Design and analysis of algorithms
 

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In this article:
1. Introduction
Part I. Low Degree Testing
2. Low Degree Testing
3. Low-degree Tests, the 99% Regime
4. Low-degree Tests, the 1% Regime
5. Gowers Norms, the Inverse Gowers Conjecture and its Failure
Part II. Higher Order Fourier Analysis
6. Nonclassical Polynomials, and the Inverse Gowers Theorem
7. Rank, Regularity, and Other Notions of Uniformity
8. Bias vs Low Rank in Large Fields
9. Decomposition Theorems
10. Homogeneous Nonclassical Polynomials
11. Complexity of Systems of Linear Forms
12. Deferred Technical Proofs
13. Algorithmic Regularity
Part III. Algebraic Property Testing
14. Algebraic Properties
15. One-Sided Algebraic Property Testing
16. Degree Structural Properties
17. Estimating the Distance from Algebraic Properties
Part IV. Open Problems
18. Open Problems
References

Abstract

Fourier analysis has been extremely useful in many areas of mathematics. In the last several decades, it has been used extensively in theoretical computer science. Higher-order Fourier analysis is an extension of the classical Fourier analysis, where one allows to generalize the “linear phases” to higher degree polynomials. It has emerged from the seminal proof of Gowers of Szemerédi’s theorem with improved quantitative bounds, and has been developed since, chiefly by the number theory community. In parallel, it has found applications also in theoretical computer science, mostly in algebraic property testing, coding theory and complexity theory. The purpose of this book is to lay the foundations of higherorder Fourier analysis, aimed towards applications in theoretical computer science with a focus on algebraic property testing.

DOI:10.1561/0400000064
ISBN: 978-1-68083-592-2
230 pp. $99.00
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ISBN: 978-1-68083-593-9
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Table of contents:
1. Introduction
Part I. Low Degree Testing
2. Low Degree Testing
3. Low-degree Tests, the 99% Regime
4. Low-degree Tests, the 1% Regime
5. Gowers Norms, the Inverse Gowers Conjecture and its Failure
Part II. Higher Order Fourier Analysis
6. Nonclassical Polynomials, and the Inverse Gowers Theorem
7. Rank, Regularity, and Other Notions of Uniformity
8. Bias vs Low Rank in Large Fields
9. Decomposition Theorems
10. Homogeneous Nonclassical Polynomials
11. Complexity of Systems of Linear Forms
12. Deferred Technical Proofs
13. Algorithmic Regularity
Part III. Algebraic Property Testing
14. Algebraic Properties
15. One-Sided Algebraic Property Testing
16. Degree Structural Properties
17. Estimating the Distance from Algebraic Properties
Part IV. Open Problems
18. Open Problems
References

Higher-order Fourier Analysis and Applications

Higher-order Fourier Analysis and Applications provides an introduction to the field of higher-order Fourier analysis with an emphasis on its applications to theoretical computer science. Higher-order Fourier analysis is an extension of the classical Fourier analysis. It has been developed by several mathematicians over the past few decades in order to study problems in an area of mathematics called additive combinatorics, which is primarily concerned with linear patterns such as arithmetic progressions in subsets of integers.

The monograph is divided into three parts: Part I discusses linearity testing and its generalization to higher degree polynomials. Part II present the fundamental results of the theory of higher-order Fourier analysis. Part III uses the tools developed in Part II to prove some general results about property testing for algebraic properties. It describes applications of the theory of higher-order Fourier analysis in theoretical computer science, and, to this end, presents the foundations of this theory through such applications; in particular to the area of property testing.

 
TCS-064