On Point Sets with Many Unit Distances in Few Directions

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 xed set of k directions taking the maximum over all sets of n points and all sets of k directions We prove that for xed k and su ciently large n n k the extremal sets are essentially sections of lattices bounded by edges parallel to the k directions and of equal length Institut f ur Informatik Freie Universit at Berlin Takustra e D Berlin Germany email brass inf fu berlin de On Point Sets with Many Unit Distances in Few Directions