On a lattice problem of H. Steinhaus

Jackson, Steve; Mauldin, R.

doi:10.1090/S0894-0347-02-00400-9

On a lattice problem of H. Steinhaus
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by Steve Jackson and R. Daniel Mauldin

J. Amer. Math. Soc. 15 (2002), 817-856

DOI: https://doi.org/10.1090/S0894-0347-02-00400-9

Published electronically: June 13, 2002

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Abstract:

It is shown that there is a subset $S$ of $\mathbb {R}^2$ such that each isometric copy of $\mathbb {Z}^2$ (the lattice points in the plane) meets $S$ in exactly one point. This provides a positive answer to a problem of H. Steinhaus.

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