224 GRADUATE STUDIES IN MATHEMATICS Discrete Analogues in Harmonic Analysis Bourgain, Stein, and Beyond Ben Krause 10.1090/gsm/224 Discrete Analogues in Harmonic Analysis Bourgain, Stein, and Beyond 224 GRADUATE STUDIES IN MATHEMATICS Discrete Analogues in Harmonic Analysis Bourgain, Stein, and Beyond Ben Krause EDITORIAL COMMITTEE Matthew Baker Marco Gualtieri Gigliola Staffilani (Chair) Jeff A. Viaclovsky Rachel Ward 2020 Mathematics Subject Classification. Primary 11-XX, 37-XX, 42-XX. For additional informationand updates on this book, visit www.ams.org/bookpages/gsm-224 Library of Congress Cataloging-in-Publication Data Names: Krause,Ben,1988-author. Title: Discreteanaloguesinharmonicanalysis: Bourgain,Stein,andbeyond/BenKrause. Description: Providence,RhodeIsland: AmericanMathematicalSociety,2022. |Series: Graduate studiesinmathematics,1065-7339;224|Includesbibliographicalreferencesandindex. Identifiers: LCCN2022024899|ISBN9781470468576(hardcover)|ISBN9781470471743(paper- back)|9781470471750(ebook) Subjects: LCSH: Harmonic analysis. | AMS: Number theory. | Dynamical systems and ergodic theory. |HarmonicanalysisonEuclideanspaces. Classification: LCCQA403.K642022|DDC515/.2433–dc23/eng20220826 LCrecordavailableathttps://lccn.loc.gov/2022024899 DOI:https://doi.org/10.1090/gsm/224 Copying and reprinting. Individual readersofthispublication,andnonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication ispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Requestsforpermission toreuseportionsofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. For moreinformation,pleasevisitwww.ams.org/publications/pubpermissions. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. (cid:2)c 2022bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttps://www.ams.org/ 10987654321 272625242322 To my family. Contents List of Symbols xi Foreword xxi Acknowledgements xxv Introduction 1 §1. Discrete Analogues in Harmonic Analysis in Action 1 §2. A Brief History of Discrete Analogues in Harmonic Analysis 3 §3. Where We’re Going 8 §4. Part One: Harmonic Analytic Preliminaries 8 §5. Part Two: Discrete Analogues in Harmonic Analysis: Radon Transforms I 9 §6. Part Three: Discrete Analogues in Harmonic Analysis: Radon Transforms II 10 §7. PartFour: DiscreteAnaloguesinHarmonicAnalysis: Maximally Modulated Singular Integrals 11 §8. Part Five: Discrete Analogues in Harmonic Analysis: An Introduction to Multilinear Theory 12 §9. Part Six: Conclusion and Appendices 12 Part 1. Harmonic Analytic Preliminaries Chapter 1. Tools 17 §1. Exploiting Invariance 18 §2. Interpolation of Lp-Spaces 23 vii viii Contents §3. The Hardy-Littlewood Maximal Function 33 §4. Continuous Operators on Infinite Dimensional Vector Spaces 47 §5. The Fourier Transform on (cid:2)2(Z) 50 §6. The Euclidean Fourier Transform 56 Chapter 2. On Oscillation and Convergence 73 Chapter 3. The Linear Theory 81 §1. The Pointwise Ergodic Theorem 81 §2. Birkhoff’s Theorem 82 §3. Introduction to Variation 91 §4. The Proof of L´epingle’s Inequality 126 Part 2. Discrete Analogues in Harmonic Analysis: Radon Transforms, I Chapter 4. Bourgain’s Maximal Functions on (cid:2)2(Z) 137 §1. Number Theoretic Approximations 142 §2. The Multi-Frequency Maximal Theory: Preliminaries 156 §3. Controling a Maximal Function on L2 161 §4. Proving the Multifrequency Maximal Theory 168 §5. Oscillation and Convergence 178 Chapter 5. Random Pointwise Ergodic Theory 183 §1. Probabilistic Preliminaries 183 §2. The Lp(X)-Theory, 1 < p < ∞ 193 §3. L1(X)-Considerations 196 Chapter 6. An Application to Discrete Ramsey Theory 219 §1. Sarko¨zy’s Theorem 220 §2. A “Pinned” Sark¨ozy Theorem 226 Chapter 7. Bourgain’s (cid:2)2(Z)-Argument, Revisited 235 §1. The Interpolative Approach 235 §2. Exploiting “Superorthogonality,” Preliminary Version 249 Part 3. Discrete Analogues in Harmonic Analysis: Radon Transforms, II Chapter 8. Ionescu-Wainger Theory 257 §1. The Introduction of Ionescu-Wainger Theory 257 Contents ix §2. Applications 263 §3. The Proof of the Variational Estimate 265 Chapter 9. Establishing Ionescu-Wainger Theory 273 §1. The Proof of the Ionescu-Wainger Construction 275 §2. Proof of the Decoupling Estimate 290 Chapter 10. The Spherical Maximal Function 301 §1. The Euclidean Theory 301 §2. The Discrete Theory 305 §3. Number Theoretic Approximations 306 §4. The Restricted Weak-Type Argument 315 Chapter 11. The Lacunary Spherical Maximal Function 319 §1. Euclidean Lacunary Averaging Operators 320 §2. The Lacunary Discrete Spherical Maximal Function 329 Chapter 12. Discrete Improving Inequalities 339 §1. Continuous Improving Inequalities: Spherical Averages 340 §2. Discrete Improving Inequalities 341 §3. Connections to Fractional Integration 354 Part 4. Discrete Analogues in Harmonic Analysis: Maximally Modulated Singular Integrals Chapter 13. Monomial “Carleson” Operators 363 §1. Introduction 363 §2. TT∗ Preliminaries and the Continuous Theory 367 §3. A Top-Down Sketch of the Argument 375 §4. Most Modulation Parameters are Safe: A TT∗ Argument 376 §5. Approximations 382 §6. Most Weyl Sums are Safe 389 §7. The Multi-Frequency Theory: Completing the Proof 393 Chapter 14. Maximally Modulated Singular Integrals: A Theorem of Stein and Wainger 403 §1. Introduction 403 §2. A Reveiw of Stein-Wainger 407 §3. Exponential Sums and Sublevel Estimates 412 §4. The Discrete Stein-Wainger Operator 417