Sets with few Intersection Numbers from Singer Subgroup Orbits

Using a Singer cycle in Desarguesian planes of order q ≡ 1(mod3), q a prime power, Brouwer 2 gave a construction of sets such that every line of the plane meets them in one of three possible intersection sizes. These intersection sizes x, y, and z satisfy the system of equations[formula]Brouwer claimed that this system has a unique solution in integers. Further, Brouwer noted that for q a perfect square, this system has a solution for which two of the variables are equal, ostensibly implying that when q is a square the constructed set has only two intersection numbers. In this paper, we perform a detailed analysis which shows that this system does not in general have a unique solution. In particular, the constructed sets when q is a square might have three intersection numbers. The cases for which this occurs are completely determined.