**The Carus Mathematical Monographs**

Volume: 37;
2021;
181 pp;
Softcover

MSC: Primary 05; 52; 11;

**Print ISBN: 978-1-4704-6031-0
Product Code: CAR/37**

List Price: $65.00

AMS Member Price: $48.75

MAA Member Price: $48.75

**Electronic ISBN: 978-1-4704-6773-9
Product Code: CAR/37.E**

List Price: $65.00

AMS Member Price: $48.75

MAA Member Price: $48.75

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#### Supplemental Materials

# The Finite Field Distance Problem

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*David J. Covert*

MAA Press: An Imprint of the American Mathematical Society

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.

#### Readership

Undergraduate and graduate students and researchers interested in combinatorics.

#### Table of Contents

# Table of Contents

## The Finite Field Distance Problem

- Cover Cover11
- Title page iii5
- Copyright iv6
- Contents vii9
- Preface ix11
- Acknowledgments xi13
- Chapter 1. Background 115
- Chapter 2. The distance problem 2135
- Chapter 3. The Iosevich-Rudnev bound 3549
- Chapter 4. Wolff’s exponent 6377
- Chapter 5. Rings and generalized distances 7589
- Chapter 6. Configurations and group actions 105119
- Chapter 7. Combinatorics in finite fields 129143
- Bibliography 169183
- Index 179193
- Back Cover Back Cover1196