For unbounded A subset of R and bounded X, the curve A times_{l1} (tX) is a geodesic ray, and d_GH(Z^n, lambda Z^n) is at least 1/2 for every lambda > 1.
Sobolev, Distribution of integer lattice points in a ball centred at a diophantine point , Mathematika 56(01):118 - 134, 2010
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New geodesic lines in the Gromov-Hausdorff class lying in the cloud of the real line
For unbounded A subset of R and bounded X, the curve A times_{l1} (tX) is a geodesic ray, and d_GH(Z^n, lambda Z^n) is at least 1/2 for every lambda > 1.