The Finite Field Distance Problem
David J. Covert · The Carus mathematical monographs · 2021
The concept of distance has been a very useful, fantastic and attractive idea for mathematicians since the period of Euclid.Paul Erdős was a well-known mathematician throughout the world for his production of conjectures and his ability to solve mathematical problems quickly.The relatively new problem "The finite field distance problem", also known as "Erdős-Falconer problem" is the seed of the well presented and well-written book by David J. Covert.This book contains 7 chapters and chapter 1 starts with the pigeon hole principle, basic algebra, finite fields and some basic inequalities.In chapter 2, the author introduces the distance problem, the Falconer distance problem and the finite field distance problem.Chapter 3 deals with the Iosevich-Rudnev bound and provides two proofs of a theorem related to the distance problem using the counting-method and L 2 -method.In chapter 4, the author discusses Wolff's "4/3-result" towards Falconer distance problem and proves a 4/3 result towards finite field distance problem.Also, in this chapter, the author provides a discussion on Fourier transform notation for finite fields.In chapter 5, the author has turned his discussions on distance problems from finite fields to finite rings and generalised distances.The author has dedicated chapter 6 to the study of configurations and group actions.Chapter 7 discusses the topics captivated by the author himself.This chapter includes incidence theory, sum-product phenomena, Kakeya conjecture, Waring's theorem and the spectral gap theorem.This is a very fantastic work of high-level scholarship and research and in level with the textbooks of I. N. Herstein, Neal H. McCoy ….I recommend this book as a textbook for graduate students and a reference book for researchers in this area of study.