Sets with Small Neighborhood in the Integer Lattice
Aaron Dall, Frederik von Heymann, Birgit Vogtenhuber · 2009
For a given cardinality we want to find lattice point configurations such that the number of lattice points with distance 1 to the set is small. Sets with the smallest number possible are called optimal. It is known that sets of points with coordinate sum less or equal to some integer k are optimal. We show that they are unique for their cardinalities. Also we will discuss the question of how to characterize optimal sets in general, and if by adding the points of distance 1 to an optimal set we will always get an optimal set. In both cases the answer is positive for the plane.