| Softcover ISBN: | 978-1-4704-6031-0 |
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| eBook ISBN: | 978-1-4704-6773-9 |
| Product Code: | CAR/37.E |
| List Price: | $65.00 |
| MAA Member Price: | $48.75 |
| AMS Member Price: | $48.75 |
| Softcover ISBN: | 978-1-4704-6031-0 |
| eBook: ISBN: | 978-1-4704-6773-9 |
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| AMS Member Price: | $97.50 $73.13 |
| Softcover ISBN: | 978-1-4704-6031-0 |
| Product Code: | CAR/37 |
| List Price: | $65.00 |
| MAA Member Price: | $48.75 |
| AMS Member Price: | $48.75 |
| eBook ISBN: | 978-1-4704-6773-9 |
| Product Code: | CAR/37.E |
| List Price: | $65.00 |
| MAA Member Price: | $48.75 |
| AMS Member Price: | $48.75 |
| Softcover ISBN: | 978-1-4704-6031-0 |
| eBook ISBN: | 978-1-4704-6773-9 |
| Product Code: | CAR/37.B |
| List Price: | $130.00 $97.50 |
| MAA Member Price: | $97.50 $73.13 |
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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, sum-product phenomena, Waring's problem, and the Kakeya conjecture.
ReadershipUndergraduate and graduate students and researchers interested in combinatorics.
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Table of Contents
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Chapters
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Background
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The distance problem
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The Iosevich-Rudnev bound
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Wolff’s exponent
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Rings and generalized distances
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Configurations and group actions
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Combinatorics in finite fields
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Additional Material
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Reviews
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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 state-of-the-art in the area; variations in the setting of, and approach to, the distance problem with explicit quantitative results; a whole chapter on the Elkes-Sharir 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 well-written 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 self-contained 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, well-illustrated figures, and curated equations.
Krystal Taylor, The Ohio State University
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RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
- Book Details
- Table of Contents
- Additional Material
- Reviews
- Requests
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.
Undergraduate and graduate students and researchers interested in combinatorics.
-
Chapters
-
Background
-
The distance problem
-
The Iosevich-Rudnev 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 state-of-the-art in the area; variations in the setting of, and approach to, the distance problem with explicit quantitative results; a whole chapter on the Elkes-Sharir 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 well-written 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 self-contained 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, well-illustrated figures, and curated equations.
Krystal Taylor, The Ohio State University
