We consider point sets in the affine plane Fq2 where each
Euclidean distance of two points is an element of Fq. These sets
are called integral point sets and were originally defined in m-dimensional
Euclidean spaces Em. We determine their maximal cardinality
I(Fq,2). For arbitrary commutative rings R
instead of Fq or for further restrictions as no three points on a
line or no four points on a circle we give partial results. Additionally we
study the geometric structure of the examples with maximum cardinality.Comment: 22 pages, 4 figure