Softcover ISBN:  9781470460310 
Product Code:  CAR/37 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
eBook ISBN:  9781470467739 
Product Code:  CAR/37.E 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
Softcover ISBN:  9781470460310 
eBook: ISBN:  9781470467739 
Product Code:  CAR/37.B 
List Price:  $130.00 $97.50 
MAA Member Price:  $97.50 $73.13 
AMS Member Price:  $97.50 $73.13 
Softcover ISBN:  9781470460310 
Product Code:  CAR/37 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
eBook ISBN:  9781470467739 
Product Code:  CAR/37.E 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
Softcover ISBN:  9781470460310 
eBook ISBN:  9781470467739 
Product Code:  CAR/37.B 
List Price:  $130.00 $97.50 
MAA Member Price:  $97.50 $73.13 
AMS Member Price:  $97.50 $73.13 

Book DetailsThe Carus Mathematical MonographsVolume: 37; 2021; 181 ppMSC: Primary 05; 52; 11
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, sumproduct phenomena, Waring's problem, and the Kakeya conjecture.
ReadershipUndergraduate and graduate students and researchers interested in combinatorics.

Table of Contents

Chapters

Background

The distance problem

The IosevichRudnev bound

Wolff’s exponent

Rings and generalized distances

Configurations and group actions

Combinatorics in finite fields


Additional Material

Reviews

This would be excellent reading material for anyone new to the area with a senior undergraduate or junior graduate background. There are a number of instructive exercises as well. The book covers the stateoftheart in the area; variations in the setting of, and approach to, the distance problem with explicit quantitative results; a whole chapter on the ElkesSharir framework, which has become one of the fundamental approaches to Erdös problems; and in the final chapter, a very nice exposition of a number of beautiful problems in the finite field model.
B. Hanson, University of Mainse 
Dave Covert's book, “The Finite Field Distance Problem”, offers a wellwritten introduction to finite field analogues of some classic Euclidean problems.
... [this title] serves as an invitation to students and mathematicians alike interested in gaining a deeper understanding of distance problems in research mathematics. While the playful narrative and selfcontained background make the book accessible to advanced undergraduates and graduate students, this book has plenty to offer professional mathematicians. The book serves as an excellent reference of breakthroughs in the field and includes many informative examples. Difficult material is made accessible with intuitive descriptions, wellillustrated figures, and curated equations.
Krystal Taylor, The Ohio State University


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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, sumproduct phenomena, Waring's problem, and the Kakeya conjecture.
Undergraduate and graduate students and researchers interested in combinatorics.

Chapters

Background

The distance problem

The IosevichRudnev bound

Wolff’s exponent

Rings and generalized distances

Configurations and group actions

Combinatorics in finite fields

This would be excellent reading material for anyone new to the area with a senior undergraduate or junior graduate background. There are a number of instructive exercises as well. The book covers the stateoftheart in the area; variations in the setting of, and approach to, the distance problem with explicit quantitative results; a whole chapter on the ElkesSharir framework, which has become one of the fundamental approaches to Erdös problems; and in the final chapter, a very nice exposition of a number of beautiful problems in the finite field model.
B. Hanson, University of Mainse 
Dave Covert's book, “The Finite Field Distance Problem”, offers a wellwritten introduction to finite field analogues of some classic Euclidean problems.
... [this title] serves as an invitation to students and mathematicians alike interested in gaining a deeper understanding of distance problems in research mathematics. While the playful narrative and selfcontained background make the book accessible to advanced undergraduates and graduate students, this book has plenty to offer professional mathematicians. The book serves as an excellent reference of breakthroughs in the field and includes many informative examples. Difficult material is made accessible with intuitive descriptions, wellillustrated figures, and curated equations.
Krystal Taylor, The Ohio State University