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STEINHAUS TILING PROBLEM AND INTEGRAL QUADRATIC FORMS
"... Abstract. A lattice L in R n is said to be equivalent to an integral lattice if there exists a real number r such that the dot product of any pair of vectors in rL is an integer. We show that if n ≥ 3 and L is equivalent to an integral lattice, then there is no measurable Steinhaus set for L, a set ..."
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Abstract. A lattice L in R n is said to be equivalent to an integral lattice if there exists a real number r such that the dot product of any pair of vectors in rL is an integer. We show that if n ≥ 3 and L is equivalent to an integral lattice, then there is no measurable Steinhaus set for L, a set which no matter how translated and rotated contains exactly one vector in L. 1.
STEINHAUS SETS AND JACKSON SETS
, 2006
"... Abstract. We prove that there does not exist a subset of the plane S that meets every isometric copy of the vertices of the unit square in exactly one point. We give a complete characterization of all three point subsets F of the reals such that there does not exists a set of reals S which meets eve ..."
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Abstract. We prove that there does not exist a subset of the plane S that meets every isometric copy of the vertices of the unit square in exactly one point. We give a complete characterization of all three point subsets F of the reals such that there does not exists a set of reals S which meets every isometric copy of F in exactly one point.
On a Lattice Problem of Steinhaus: Simultaneous tilings of the Plane
, 2013
"... (1950’s):(1) Are there subsets A, S ⊂ R 2 such that card(S ∩ T (A)) = S ∩ T (A)  = 1, for all isometries T of R 2? Let’s call such a set S a Steinhaus set for A. A = R2 and S  = 1 is ruled out. The trivial case where T.: Sierpinski (1958), Erdös (1985) Yes. Steinhaus (1950’s): What if the set ..."
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(1950’s):(1) Are there subsets A, S ⊂ R 2 such that card(S ∩ T (A)) = S ∩ T (A)  = 1, for all isometries T of R 2? Let’s call such a set S a Steinhaus set for A. A = R2 and S  = 1 is ruled out. The trivial case where T.: Sierpinski (1958), Erdös (1985) Yes. Steinhaus (1950’s): What if the set A is specified? In particular, A = Z or A = Z2, i.e., Can S be a fundamental domain simultaneously for all rotations of Z2? Steinhaus ’ questions appeared in Sierpinski’s 1958 paper on this subject. T.: Komjath (1992) S exists if A = Z or A = Q × Q. The problem remained: what if A = Z 2? To get a feeling for the problem, let’s check some other dimensions. 2 T.: There is a Borel set which is a Steinhaus set for Z 1 in R 1, namely