Dimensions of strong \(n\)-point sets (Q818381)

An \(n\)-point set is a set in the plane which hits each line in exactly \(n\) points, and a strong \(n\)-point set hits each line and each circle in exactly \(n\) points. Since the notation of \(n\)-point set was introduced by \textit{S. Marzukiewicz} in 1914 [Travaux de topologie et ses applications, PWN-Éditions Scientifiques de Pologne (1969; Zbl 0202.53501)], it is known that \(n\)-point sets exist for each \(n\), but the constructions are not explicit. The notion of strong \(n\)-point set was introduced by \textit{K. Bouhjar, J. J. Dijkstra} and \textit{J. van Mill} [Topology Appl. 112, 215--227 (2001; Zbl 1021.54010)]. Thereby properties of those sets are not apparent. However several results on dimensions have recently been made. \textit{J. Kuesza} [Proc. Am. Math. Soc. 116, 551--553 (1992; Zbl 0765.54006 )], \textit{D. L. Fearnley, L. Fearnley} and \textit{J. W. Lamoreaux} [Proc. Am. Math. Soc. 131, 2241--2245 (2003; Zbl 1023.54014)] and \textit{K. Bouhjar, J. J. Djikstra} and \textit{J. van Mill} [loc. cit.] showed that every \(2\)-point set, \(3\)-point set and strong \(3\)-point set is zero-dimensional. On the other hand, \textit{K. Bouhjar, J. J. Djikstra} and \textit{J. van Mill} [loc. cit.] also proved that for all \(n \geq 4\) there exists a one-dimensional \(n\)-point set. In this paper the author shows that all strong \(4\)- and \(5\)-point sets are zero-dimensional and for all \(n \geq 6\) both zero-dimensional and one-dimensional strong \(n\)-point sets exist.