A study by Elkies and McMullen in 2004 showed that the gaps between the fractional parts of √ n for n = 1, . . . , N, have a limit distribution as N tends to infinity. The limit distribution is non-standard and differs distinctly from the exponential distribution expected for independent, uniformly distributed random variables on the unit interval. We complement this result by proving that the two-point correlation function of the above sequence converges to a limit, which in fact coincides with the answer for independent random variables. We also establish the convergence of moments for the probability of finding r points in a randomly shifted interval of size 1/N . The key ingredient in the proofs is a non-divergence estimate for translates of certain non-linear horocycles.

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 [Annals of Math. 173 (2010Math. 173 ( ), 1949Math. 173 ( -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 markedly from the gap distribution observed in a Poisson process, which is exponential. The present study shows that the limiting two-point correlation function of the projected lattice points exists and is Poissonian. This answers a recent question by Boca, Popa and Zaharescu [arXiv:1302.5067]. The existence of the limit is subject to a certain Diophantine condition. We also establish the convergence of more general mixed moments.

Translation of the Bible or any other text unavoidably involves a determination about its meaning. There have been different views of meaning from ancient times up to the present, and a particularly Enlightenment and Modernist view is that the meaning of a text amounts to whatever the original author of the text intended it to be. This article analyzes the authorial-intent view of meaning in comparison with other models of literary and legal interpretation. Texts are anchors to interpretation but are subject to individualized interpretations. It is texts that are translated, not intentions. The challenge to the translator is to negotiate the meaning of a text and try to choose the most salient and appropriate interpretation as a basis for bringing the text to a new audience through translation.

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Let Γ < PSL(2, C) be a geometrically finite non-elementary discrete subgroup, and let its critical exponent δ be greater than 1. We use representation theory of PSL(2, C) to prove an effective bisector counting theorem for Γ, which allows counting the number of points of Γ in general expanding regions in PSL(2, C) and provides an explicit error term. We apply this theorem to give power savings in the Apollonian circle packing problem and related counting problems.