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ABSTRACT

Herr and Kwak had recently established sharp estimates for certain exponential sums on the 3 dimensional torus by counting rectangles in the plane. I will try to explain their work and highlight connections to Fourier analysis. One interesting question we will address is the following. Given N points in the plane, how many rectangles can we form so that all vertices are in the given points? Surprisingly, the answer has to do with the topology of $\mathbb{R}^2$, and is the key to unlocking the aforementioned exponential sum estimates. These exponential estimates in turn provide sharp bounds for solutions to the periodic Schrodinger equation in 2+1 dimensions.