Delone Sets Generated by Square Roots

Delone sets are locally finite point sets, such that (a) any two points are separated by a given minimum distance, and (b) there is a given radius so that every ball of that radius contains at least one point. Important examples include the vertex set of Penrose tilings and other regular model sets, which serve as a mathematical model for quasicrystals. In this note, we show that the point set given by the valuesn e2 pi i alpha nwithn=1,2,3, horizontal ellipsis is a Delone set in the complex plane, for any alpha>0. This complements Akiyama's recent observation (see Spiral Delone sets and three distance theorem (2020),Nonlinearilty, 33(5): 2533-2540) thatn e2 pi i alpha nwithn=1,2,3, horizontal ellipsis forms a Delone set, if and only if alpha is badly approximated by rationals. A key difference is that our setting does not require Diophantine conditions on alpha.