One of the central aims in quantum chaos is to classify quantum systems according to universal statistical properties. It has been conjectured that the energy levels of generic integrable quantum systems have the same statistical properties as random numbers from a Poisson process (Berry & Tabor 1977) , and chaotic quantum systems the same as eigenvalues of random matrices from suitably chosen ensembles (Bohigas, Giannoni & Schmit 1984). I review some recent developments concerning simple

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... s of integrable systems, where the study of eigenvalue correlations leads to subtle lattice point counting problems which, in some instances, can be solved by ergodic theoretic techniques. In a special example (the so-called "boxed oscillator") energy level statistics are related to the statistical distribution of the fractional parts of the sequence n 2 α. We will see that the error term of this distribution can be identified with an almost modular function, and that the fluctuations of the error term are governed by a general limit theorem for such functions.