Erdos 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.
General
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