We will discuss the fine-scale distribution of rational points in n dimensions, with denominators restricted to a window at large height but of relatively small width. Previous results in the literature, due to Hall and others, were limited to Farey sequences, where the window size is of the same order as the denominator. We prove the convergence of fine-scale statistics in a range of scaling limits and express the limit laws in terms of natural probability measures on the space of affine lattices. The key technical ingredient of our approach is an equidistribution theorem for slowly expanding horospheres, with some new exotic limit measures. Our techniques furthermore allow us to answer a recent question by Anderson, Boca, Cobeli and Zaharescu concerning the directional statistics of lattice points. The talk is based on joint work with Gaurav Aggarwal and Anish Ghosh.