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 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 > n ₀ (k) , the extremal sets are essentially sections of lattices, bounded by edges parallel to the k directions and of equal length. <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p355.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received January 10, 1997, and in revised form May 16, 1997.