Superoscillations, the Talbot Carpet, and Gauss Sums.
Superoscillations are an interesting phenomenon that appears in different areas, including optics and weak quantum measurements. An important question is whether a superoscillating function maintains its superoscillatory nature when evolved according to specific forms of the Schrodinger equation. In this talk we will consider the evolution of the Dirac comb (infinite sum of deltas) to show how to recover, optically, the generalized quadratic Gauss sums. We will then use the theory of superoscillations to show how such Gauss sums can be asymptotically recovered from the values of the spectrum of any sufficiently regular function with compact support.