The Finite Field Distance Problem (Carus Mathematical Monographs) (The Carus Mathematical Monographs, 37)

Erdős asked how many distinct distances must there be in a set of $n$ points in the plane. Falconer asked a continuous analogue, essentially asking what is the minimal Hausdorff dimension required of a compact set in order to guarantee that the set of distinct distances has positive Lebesgue measure in $R$. The finite field distance problem poses the analogous question in a vector space over a finite field. The problem is relatively new but remains tantalizingly out of reach. This book provides an accessible, exciting summary of known results. The tools used range over combinatorics, number theory, analysis, and algebra. The intended audience is graduate students and advanced undergraduates interested in investigating the unknown dimensions of the problem. Results available until now only in the research literature are clearly explained and beautifully motivated. A concluding chapter opens up connections to related topics in combinatorics and number theory: incidence theory, sum-product phenomena, Waring's problem, and the Kakeya conjecture. Cover Title page Copyright Contents Preface Acknowledgments Chapter 1. Background 1.1. Equivalence relations and the pigeonhole principle 1.2. Algebra and finite fields 1.3. Basic inequalities 1.4. Notation 1.5. Exercises: Chapter 1 Chapter 2. The distance problem 2.1. Introduction to the distance problem 2.2. Falconer distance problem 2.3. Finite field distance problem 2.4. Exercises: Chapter 2 Chapter 3. The Iosevich-Rudnev bound 3.1. Counting-method 3.2. The L2-method 3.3. Finite field spherical averages 3.4. Size and decay estimates for spheres 3.5. Finite field counterexample 3.6. Relations to the Falconer problem 3.7. Exercises: Chapter 3 Chapter 4. Wolff’s exponent 4.1. Introduction 4.2. Proof of L2 estimate for ν(t) 4.3. Restriction and extension theory 4.4. Exercises: Chapter 4 Chapter 5. Rings and generalized distances 5.1. Distances in finite rings 5.2. Distances between two sets 5.3. Generalized distances 5.4. Pinned distances 5.5. Exercises: Chapter 5 Chapter 6. Configurations and group actions 6.1. Finite configurations 6.2. The Elekes-Sharir framework 6.3. Triangles: The enquote {7/4} bound 6.4. Triangles: The enquote {8/5} bound 6.5. Distance graph 6.6. Exercises: Chapter 6 Chapter 7. Combinatorics in finite fields 7.1. Incidence theory 7.2. Sum-product phenomena 7.3. Kakeya conjecture 7.4. Waring’s theorem 7.5. Roth’s theorem and the cap-set problem 7.6. The spectral gap theorem 7.7. Exercises: Chapter 7 Bibliography Index Back Cover