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The Erdős Distance Problem

Julia Garibaldi, Alex Iosevich and Steven Senger
American Mathematical Society
Publication Date: 
Number of Pages: 
Student Mathematical Library 56
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Charles Ashbacher
, on

The majesty, beauty and mystery of mathematics is largely developed from the fact that a single problem generally has so many consequences and paths that can branch off into applications or theoretical consequences of the basic problem. The original Erdős distance problem is simple to state:

What is the smallest number of distinct distances between points of a large finite subset of the Euclidean space of dimension two?

Once the initial problem is studied, there is the logical extension into spaces with dimensions greater than two, altered definitions of the term distance using other metrics (for example the potato metric) and some applications.

The treatment is a bit difficult at times, but the advanced and determined undergraduate will have no trouble understanding it. Some graph theory is covered, as that is a logical place to go with a set of points in a Euclidean space. The most extensive application examined is information theory, one of the most important applications of mathematics in the digital age.

A large number of exercises are included, although solutions are not. This book would make an excellent text for a special topics course or colloquium project as it demonstrates how mathematics starts with one relatively simple idea and then moves like a wayward cancer cell in many initially unknown and sometimes unexpected directions.

Charles Ashbacher splits his time between consulting with industry in projects involving math and computers, teaching college classes and co-editing The Journal of Recreational Mathematics. In his spare time, he reads about these things and helps his daughter in her lawn care business.

  • Introduction
  • The $\sqrt{n}$ theory
  • The $n^{2/3}$ theory
  • The Cauchy-Schwarz inequality
  • Graph theory and incidences
  • The $n^{4/5}$ theory
  • The $n^{6/7}$ theory
  • Beyond $n^{6/7}$
  • Information theory
  • Dot products
  • Vector spaces over finite fields
  • Distances in vector spaces over finite fields
  • Applications of the Erdős distance problem
  • Hyperbolas in the plane
  • Basic probability theory
  • Jensen's inequality
  • Bibliography
  • Biographical information
  • Index of terminology