# On point sets with many unit distances in few directions

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Peter Braß

Graduiertenkolleg Algorithmische Diskrete Mathematik

Institut für Informatik

Freie Universität Berlin

Takustr. 9, D-14195 Berlin, Germany

email: brass@inf.fu-berlin.de

Report B 97-04

June 1997

We study the problem
of the maximum number of unit distances among n points
in the plane under the additional restriction that we count
only those unit distances that occur in a fixed set of k directions,
taking the maximum over all sets of n points and all sets of
k directions.
We prove that for fixed k and sufficiently large n
(depending on k) the
extremal sets are essentially sections of lattices, bounded
by edges parallel to the k directions and of equal length.

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