Proof of the Finiteness of the Chromatic Number of Two-Dimensional Lacunary Distance Graphs

We extend the one-dimensional lonely set method to two dimensions for the purpose of studying the chromatic number of integer distance graphs in two dimensions. Given a lacunary sequence of displacement vectors in $Z^2$, we use a lacunary matrix theorem given by Broderick, Fishman and Kleinbock, to prove the existence of a satisfactory multiplier vector. We then give an explicit geometric colouring argument. This proves that any integer distance graph generated by a lacunary sequence of vectors in two dimensions has finite chromatic number.