On the number of directions determined by a set of points in an affine Galois plane
In \(AG (2,q)\), let \(U\) be a set of \(q-n\) points. The author studies the set of ``points at infinity'' \((m)\) such that \(m\) is the slope of some line \(PQ\) where \(P\) and \(Q\) are distinct points of \(U\). Let \(N\) be the number of such points at infinity. Let \(D\) be a set of points at infinity and let \(W\) be a set of affine points. Then \(W\) is said to be a \(D\)-set if \(w\) determines precisely the points of \(D\). The \(D\)-set \(W\) is complete (with respect to \(D)\) if \(W\) is maximal. The main result: If \(U\) is a \(D\)-set with \(q - n\) points, \(n \leq {1 \over 2} \sqrt q\) and if \(|D |< {1 \over 2} (q + 1)\) then \(U\) is incomplete.
scientific article; zbMATH DE number 883806