Small almost complete arcs (Q1849907)

A \(k\)-arc in a projective plane is a set of \(k\) points no three of which are collinear. An arc is complete if it is not properly contained in another arc. \textit{L. Lunelli} and \textit{M. Sce} [Ist. Lombardo Accad. Sci. Lett., Rend., Sci. Mat. Fis. Chim. Geol., Ser. A 98, 3-52 (1964; Zbl 0131.36802)] proved that if \(K\) is a complete \(k\)-arc in a projective plane of order \(q\), then \(k > \frac{1}{2}(3 + \sqrt{8q+1}) \approx \sqrt{2q}\). A \(k\)-arc is complete if and only if every point of the plane is contained in at least one secant of the arc. The author constructs a sequence of \(k\)-arcs contained in irreducible conics of desarguesian planes of order \(q\), for some odd prime powers \(q\), such that \(k \lesssim 13(\log_2 q^2) \sqrt{q}\) and such that, if \(q\) goes to infinity, the ratio of the number of points not on a secant of the \(k\)-arc to the total number of points in the plane, goes to 0.