On distance sets in the triangular lattice (Q2799704)

The paper studies the distinct distances problem of \textit{P. Erdős} [Am. Math. Mon. 53, 248--250 (1946; Zbl 0060.34805)] in a particular discrete setting, when points in the plane are points of a triangular lattice within a certain triangle. This setting was first investigated by the first author and \textit{H. Snevily} [Electron. J. Comb. 20, No. 4, Research Paper P33, 13 p. (2013; Zbl 1295.52011)]. They conjectured that if a distance occurs between two points, then it also occurs between two boundary points. This conjecture is proved in the paper under review. The paper counts exactly and asymptotically the number of distinct distances among the points in the setting above. The asymptotic results are analogous to the asymptotic results in the construction of Erdős based on points of a square lattice.