Distribution of error terms in lattice points counting

Faculty Mentor: Junxian Li

Description: Given a nice convex region \(\Omega\) and a large number \(x\), we know that the number of lattice points in \(x\Omega\) is asymptotically the volume of the region \(x\Omega\). How does the error term in the approximation behave as \(x\) gets larger? What is the distribution of the error term as we vary \(x\) in a large interval? If the region is a disk, then this related to the Gauss circle problem, which gives an upper bound for the error term. The normalized error term is known to have a non-gaussian distribution. In this project, we will explore the distribution of the error term in these lattice point counting problems for general ellipsoids in higher dimensions.

Requirements: Real and complex analysis or basic coding skills.

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