Proves finite chromatic number for any 2D lacunary integer distance graph in Z^2 by extending the lonely set method via Broderick-Fishman-Kleinbock theorem and explicit geometric coloring.
Dubickas,On the fractional parts of lacunary sequences, Mathematica Scandinavica, 99(1) (2006), 136–146
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Proof of the Finiteness of the Chromatic Number of Two-Dimensional Lacunary Distance Graphs
Proves finite chromatic number for any 2D lacunary integer distance graph in Z^2 by extending the lonely set method via Broderick-Fishman-Kleinbock theorem and explicit geometric coloring.