ENGLISH

The Finite Field Distance Problem (Carus Mathematical Monographs) (The Carus Mathematical Monographs, 37)

By David J. Covert (author)

Book information

Publisher
American Mathematical Society
Year
2021
ISBN
1470460319, 9781470460310
Language
english
Format
PDF
Filesize
3 MB (3272818 bytes)
Pages
181\196
Time added
2023-09-03 08:21:07

Description

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.

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