On cyclic \(k\)-arcs of Singer type in \(\text{PG}(2,q)\) (Q1849888)

A \(k\)-arc in a projective plane is a set of points, no three of which are collinear. Suppose now that \(K\) is a \(k\)-arc in \(\text{PG}(2,q)\), \(q=p^l\), \(p\) prime, consisting of the points of a point orbit under a subgroup \(G\) of a Singer group of \(\text{PG}(2,q)\). The author proves that if \(p\) is greater than 5, and if \(2k\) is different from \(-2,1,2,4 \pmod p\), then \(k \leq \frac{44}{45}q+\frac{8}{9}\).