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We prove the convergence of moments of the number of directions of affinelattice vectors that fall into a small disc, under natural Diophantineconditions on the shift. Furthermore, we show that the pair correlationfunction is Poissonian for any irrational shift in dimension 3 and higher,including well-approximable vectors. Convergence in distribution was alreadyproved in the work of Str\"ombergsson and the second author, and the principalstep in the extension to convergence of moments is an escape of mass estimatefor averages over embedded $\operatorname{SL}(d,\mathbb{R})$-horospheres in thespace of affine lattices.

Poissonian pair correlation for directions in multi-dimensional affine lattices, and escape of mass estimates for embedded horospheres