In these notes we discuss the self-reducibility property of the Weil representation. We explain how to use this property to obtain sharp estimates of certain higher-dimensional exponential sums which originate from the theory of quantum chaos. As a r
esult, we obtain the Hecke quantum unique ergodicity theorem for generic linear symplectomorphism $A$ of the torus $T^{2N}=R^{2N}/Z^{2N}.
The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors for the DFT. The tr
ansition matrix from the standard basis to the canonical basis defines a novel transform which we call the discrete oscillator transform (DOT for short). Finally, we describe a fast algorithm for computing the discrete oscillator transform in certain cases.
Modifications to gravity that add additional functions of the Ricci curvature to the Einstein-Hilbert action -- collectively known as $f(R)$ theories -- have been studied in great detail. When considered as complete theories of gravity they can gener
ate non-perturbative deviations from the general relativistic predictions in the solar system, and the simplest models show instabilites on cosmological scales. Here we show that it is possible to treat $f(R)=Rpmmu^4/R$ gravity in a perturbative fashion such that it shows no instabilities on cosmological scales and, in the solar system, is consistent with measurements of the PPN parameters. We show that such a theory produces a spatially flat, accelerating universe, even in the absence of dark energy and when the matter density is too small to close the universe in the general relativistic case.
