In this paper we study the number of special directions of sets of cardinality divisible by p on a finite plane of order p , where p is a prime. We show that there is no such a set with exactly two special directions. We characterise sets with exactly three special directions which answers a question of Ghidelli in negative. Further we introduce methods to construct sets of minimal cardinality that have exactly four special directions for small values of p .
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Kiss et al. (2024) studied this question.
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