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We consider an affine Euclidean lattice and record the directions of all lattice vectors of length at most T. Strömbergsson and the second author proved in [Ann. Math. 173 (2010), 1949–2033] that the distribution of gaps between the lattice directions has a limit as T tends to infinity. For a typical affine lattice, the limiting gap distribution is universal and has a heavy tail; it differs marked...

The Distribution of Directions in an Affine Lattice: Two-Point Correlations and Mixed Moments