This electronic thesis or dissertation has been downloaded from Explore Bristol Research, http://research-information.bristol.ac.uk Author: Stevens, Sophie C C Title: Incidence Geometry in the Plane and Applications to Arithmetic Combinatorics General rights Access to the thesis is subject to the Creative Commons Attribution - NonCommercial-No Derivatives 4.0 International Public License. A copy of this may be found at https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode This license sets out your rights and the restrictions that apply to your access to the thesis so it is important you read this before proceeding. Take down policy Some pages of this thesis may have been removed for copyright restrictions prior to having it been deposited in Explore Bristol Research. However, if you have discovered material within the thesis that you consider to be unlawful e.g. breaches of copyright (either yours or that of a third party) or any other law, including but not limited to those relating to patent, trademark, confidentiality, data protection, obscenity, defamation, libel, then please contact [email protected] and include the following information in your message: •Your contact details •Bibliographic details for the item, including a URL •An outline nature of the complaint Your claim will be investigated and, where appropriate, the item in question will be removed from public view as soon as possible. Incidence Geometry in the Plane and Applications to Arithmetic Combinatorics Sophie Clara Charlotte Stevens A dissertation submitted to the University of Bristol in accordance with the requirements for award of the degree of Doctor of Philosophy in the Faculty of Science. School of Mathematics, 12th December 2019 Word count: 34,000 Abstract In this thesis we study the applications of incidence geometry to additive combinatorics, with a particular focus of the setting of the prime residue field. We begin by introducing some context to incidence geometry and its role in combinatorics. In Chapters 4 and 5, we prove a new incidence bound between points and lines in the plane and discuss its applications. The sum-product phenomenon is a motivating problem of this thesis and we discuss this in both the case of an arbitrary field and the specific case of F = R. In this latter setting, in Chapter 6 we prove a bound on the energy formulation of the sum-product problem. A consequence of this is to the sum-product problem itself, as well as to that of expander functions. Finally, in Chapter 8, we prove a new bound on the number of pinned distances in the plane. iii Acknowledgements This document could not and did not happen in isolation and this page presents a chance to begin to express my thanks towards those who have helped me produce it. First and foremost I am indebted to my supervisor Misha Rudnev, for his wisdom, his patience and his kindness. Thank you to Lynne Walling for her accuracy in recommending him as a Good Person, and also for encouraging me to do a PhD. Thanks to Julia Wolf, who provided me with countless opportunities, and to Olly Roche-Newton and Sean Prendiville who are doubtless unaware of the encouragement they gave me. I am grateful of the opportunities that the HARICOT and SPACE seminars provided, and to the combinatorics department at Bristol who provided the material and companionship. I have learnt lots from working with Brendan Murphy, and also with Frank de Zeeuw and Ilya Shkredov. I am grateful to Jozsef Solymosi, especially for the hospitality he provided at the University of British Columbia. My time at Bristol has been greatly enhanced by the people I have met who have kept me sane and happy. Ana Costache, Nick Jones and Luka Rimani´c are the three who have played the greatest role in this task, but Justin D., Charley, Emma, Jenny, Simon, Lydia and Olly K. have all played their parts in my contentment. For those I have have known the longest and who have given me support in all parts of my life, it gives me great pleasure to thank Kats, Cate, Robert and the rest of my (extended) family, but most of all, Tim. v Author's declaration I declare that the work in this dissertation was carried out in accordance with the requirements of the University's Regulations and Code of Practice for Research Degree Programmes and that it has not been submitted for any other academic award. Except where indicated by specific reference in the text, the work is the candidate's own work. Work done in collaboration with, or with the assistance of, others, is indicated as such. Any views expressed in the dissertation are those of the author. S.C.C. Stevens December 12th, 2019 vii Publications Some parts of this thesis appear in the following publications: • [108] An Improved Point-Line Incidence Bound over Arbitrary Fields joint work with Frank de Zeeuw Bulletin of the London Mathematical Society 49.5 (2017), pp. 842-858 • [92] On the Energy Variant of the Sum-Product Conjecture joint work with Misha Rudnev and Ilya Shkredov To appear: Revista Matem´aticaIberoamericana, Electronically published on September 10, 2019 • [73] Bisector Energy and Pinned Distances in Positive Characteristic joint work with Brendan Murphy and Misha Rudnev arXiv preprint: 1908.04618 Sections containing reproduction are clearly indicated in the text. ix Contents 1 Introduction 1 2 A beginner's guide to incidence theorems 5 2.1 Incidences in two dimensions . 5 First incidence bounds . 7 The Szemer´edi-Trotter Theorem . 8 The number of k-rich points and lines . 10 Incidences over R3 ......................... 11 2.2 Incidences in arbitrary fields . 11 How is F different to R?...................... 11 Incidences over F2 ......................... 12 Large sets of points . 13 Small sets of points . 13 Incidences over F3 ......................... 13 Points-Lines . 13 Points-Planes . 14 2.3 Collinear triples . 16 2.4 Addendum: Projective transformations . 18 3 A beginner's guide to arithmetic structure 21 3.1 Arithmetic Structure in a Set and the Sum-Product Conjecture 21 Quantifying Structure: Cardinality . 22 Progress on the Sum-Product Phenomenon . 24 Addendum: The Pl¨unnecke-Ruzsa Inequality . 25 x CONTENTS 3.2 Arithmetic Energy . 26 Definitions and Examples . 26 Higher energies . 27 Geometric interpretation of energy . 28 3.3 How Cardinality and Energy Interact . 30 Cauchy-Schwarz inequality . 30 Addendum: The Balog-Szemer´edi-Gowers Theorem . 31 Solymosi's Approach to the Sum-Product Problem . 32 3.4 Other approaches to the sum-product phenomenon . 34 Few sums or few products . 34 On the size of jAA + Aj or jA(A + A)j . 35 4 Incidences between points and lines in arbitrary fields 37 4.1 Introduction . 37 4.2 Incidences over R .......................... 38 4.3 Incidences in Finite Fields . 39 4.4 Main Results . 40 4.5 Discussion . 41 Optimality . 41 Bounding rich points and lines . 42 When to use Theorems 4.4 and 4.5? . 44 4.6 Proof of Theorem 4.5: Cartesian incidence bound . 45 4.7 Proof of Theorem 4.4 . 47 Proof of Theorem 4.4 . 49 4.8 Open questions . 51 5 Applications of point-line incidences 53 5.1 Introduction . 53 5.2 Set expansion and sum-product estimates . 53 The Sum-Product Phenomenon . 53 Sum-Product over F .................... 54 Variations of the sum-product problem . 56 5.3 Geometric applications . 58 Distinct distances and the pinned distance problem . 58 Beck's theorem . 62 5.4 Collinear Quadruples . 65 5.5 Open Questions . 66 xi CONTENTS 6 Energy decomposition of a set 67 6.1 Introduction . 67 Structure of this chapter . 68 6.2 On the energy formulation and Balog{Wooley decomposition . 68 Balog{Wooley decomposition . 70 Main Decomposition Results . 71 6.3 Proof of Decomposition Results . 72 Initial Decomposition . 72 Bootstrapping the initial decomposition and proofs of Theo- rems 6.5 and 6.6 . 79 Proof of Theorems 6.5 and 6.6 . 80 Proof of Corollary 6.7 . 82 7 Applications of energy decomposition 83 7.1 Road-map of this chapter . 83 7.2 Application: Expansion . 83 Expander functions . 83 A new bound on a four-variable expander . 85 Proof of Theorem 7.2 . 85 7.3 The Sum Product Phenomenon . 91 Subsequent Improvements . 92 Proof of Theorem 7.8 . 93 Proof of Theorem 7.8 . 93 Proving Theorem 7.10 . 97 Proof of Theorem 7.10 . 100 Conclusion of the proof of Theorem 7.10 . 102 8 Pinned distances in positive characteristic 103 8.1 Introduction . 103 Structure of this chapter . 104 8.2 Distinct distances over F . 105 What changes? . 105 The case of large sets . 107 Trivial distance bound . 108 8.3 The pinned distance problem . 109 8.4 Literature review . 110 The distinct distances problem over R . 110 xii CONTENTS The Elekes{Sharir Framework . 111 Related work over F . 111 8.5 Main Results . 112 Sub-optimal Result . 113 8.6 Discussion of techniques . 113 High-level overview of pinned distances strategy . 113 Pinned distance strategy: perpendicular bisectors . 114 Counting perpendicular bisectors with incidences . 116 8.7 A toolkit for distinct distances . 117 A framework for distance preserving transformations . 117 Blaschke-Gr¨unwald Kinematic Mapping . 119 Isotropic lines and Perpendicular Bisectors . 120 Axial Symmetries . 121 Incidence Geometry . 123 8.8 Proof of Theorems 8.10 and 8.11 . 124 From pinned distances to isosceles triangles . 124 Bounding isosceles triangles . 126 Count of isosceles triangles with incidence geometry . 127 Proof of Theorem 8.11 . 128 Proof of Theorem 8.10 . 129 8.9 Proof of Proposition 8.12 . 130 Case 1: low multiplicity . 131 Case 2: high multiplicity . 132 8.10 Future work .