Sets meeting isometric copies of the lattice Z2 in exactly one point

摘要

The construction of a subset S of ℝ² such that each isometric copy of ℤ² (the lattice points in the plane) meets S in exactly one point is indicated. This provides a positive answer to a problem of H. Steinhaus [Sierpiński, W. (1958) Fund. Math. 46, 191–194].