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Abstract
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Let
be an arbitrary
finite field, and
be a point set in
.
Let
be the set of distances determined by pairs of points in
.
Using Kloosterman sums, Iosevich and Rudnev (2007) proved that if
then
. In
general, this result is sharp in odd-dimensional spaces over arbitrary finite fields.
We use the point-plane incidence bound due to Rudnev to prove that if
has
Cartesian product structure in vector spaces over prime fields, then we can break the
exponent
and still cover all distances. We also show that the number of pairs of points in
of
any given distance is close to its expected value.
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Keywords
distances, finite fields, incidence, Rudnev's point-plane
incidence bound
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Mathematical Subject Classification
Primary: 14N10, 51A45, 52C10
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Milestones
Received: 18 June 2020
Revised: 22 July 2020
Accepted: 29 July 2020
Published: 14 January 2021
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