  There are many examples for point sets in finite geometry, which behave
"almost regularly" in some (well-defined) sense, for instance they have "almost
regular" line-intersection numbers. In this paper we investigate point sets of
a desarguesian affine plane, for which there exist some (sometimes: many)
parallel classes of lines, such that almost all lines of one parallel class
intersect our set in the same number of points (possibly mod $p$, the
characteristic). The lines with exceptional intersection numbers are called
renitent, and we prove results on the (regular) behaviour of these renitent
lines.
