Sets of type \((M,N) \bmod q\) with respect to hyperplanes in \(PG(r,q)\) (Q1339787)

The author studies the sets \(S\) of \(\text{PG}(r,q)\) \((r,q \geq 2)\) such that for every hyperplane \(\Pi\), \(| S\cap \Pi | \equiv M,N \pmod q\), with \(N-M\) coprime of \(q\), that is of type \((M,N)\) respect to hyperplanes. He proves that if the cardinality of \(S\) is congruent either \(M\) or \(N \bmod q\), then the hyperplanes such that \(| S \cap \Pi| \equiv M \pmod q\) (or \(\equiv N \bmod q\)) are incident with a fixed point \(X\) (or \(Y\)). If the cardinality of \(| S\cap \Pi|\) is equal to \(M\) or congruent to \(N\pmod q\), where \(0 \leq M \leq r - 2\), than the cardinality of \(S\) is not greater than \(q^ M + M\), generalizing a result of A. Blokhuis and F. Mazzocca. Then he gives some applications of the previous results to \(\text{AG}(r,q)\), \(\text{PG}(r,2)\) and a class of finite linear spaces.