Lattice points in a rectangle
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چکیده
Basically, what we can try to do is to take the Fourier transform of 1R (the indicator function for the set R), and then use Fourier inversion to show that for each shift t+L of the lattice L, this shift intersects the set R. Actually, it will not quite be enough to take the Fourier transform of just 1R, but instead we will need to work with a smoothed version of 1R in order to make the Fourier transform have the requisite “decay properties”. Basically, we will need that the Fourier transform of our function is in L. It turns out that the dual group R̂ is isomorphic to R, and that the additive characters χ : R → C take the form χ(~x) = e. Fourier transforms are then defined via
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