A problem of Steinhaus concerning the existence of a plane set with a certain property (Q2707560)

A subset of \(\mathbb{R}^2\) is said to satisfy the Steinhaus property if under any rigid motion of \(\mathbb{R}^2\), it contains exactly one integer lattice point. The major open problem is whether or not such a set exists and it is one of the many interesting questions involving integer lattice points in the plane. For reference to these, see [Lattice points (Pitman, Harlow 1989; Zbl 0683.10025)] by \textit{P. Erdős, P. M. Gruber} and \textit{J. Hammer}. The problem of Steinhaus has also been mentioned in [``Research problems in discrete geometry'', 6th ed. (1981; Zbl 0528.52001)] by \textit{W. Moser}. NEWLINENEWLINENEWLINEIn this paper the authors describe various partial results obtained so far in this direction and few other closely related problems of Steinhaus.NEWLINENEWLINEFor the entire collection see [Zbl 0932.00040].