@MISC{Jackson_ona, author = {Steve Jackson and R. Daniel Mauldin}, title = {On a Lattice Problem of H. Steinhaus}, year = {} }

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Abstract

. It is shown that there is a subset S of R 2 such that each isometric copy of 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. 1. Introduction Sometime in the 1950's, Steinhaus posed the following problem. Do there exist two sets A and B in the plane such that every set congruent to A has exactly one point in common with B? The trivial case where one of the sets is the plane and the other consists of a single point is ruled out. The rst appearance of this problem in the literature seems to be in a 1958 paper of Sierpinski [11]. In this paper, he showed the answer is yes, a result later rediscovered by Erd}os [5]. Of course, there are many variants of this problem. For example, one could specify the set A. In this direction, Komjath showed that such a set exists if A = Z, the set of all integers [10]. Steinhaus also asked about the specic case where A = Z 2 . The rst reference to this problem a...