STNB2017, (31st edition)
Lattice points in elliptic paraboloids
Presenters
Carlos Pastor
Abstract
Given a compact subset $\mathcal{K}$ of $\mathbb{R}^d$, let $\mathcal{N}(R)$ denote the number of points in the lattice $\mathbb{Z}^d$ that lie within $\mathcal{K}$ after being dilated by a factor $R > 1$. The lattice point problem associated to $\mathcal{K}$ consists in approximating $\mathcal{N}(R)$ for large $R$, usually in the form $\mathcal{N}(R) = |\mathcal{K}|R^d + O(R^\alpha)$ where $|\mathcal{K}|$ stands for the volume of $\mathcal{K}$. Determining how small the exponent $\alpha$ can be taken in this asymptotic formula is an open problem even for simple regions such as the circle (Gauss problem) or rational ellipsoids in the three-dimensional space. In this talk I will present a recent joint work with F. Chamizo, showing that $\alpha$ may be taken arbitrarily close to $d-2$ for a wide family of truncated elliptic paraboloids in at least three dimensions, and certainly not smaller than $d-2$ in some cases. To do this the problem is first reduced to the estimation of a 2-dimensional quadratic exponential sum, which is then bounded using a simplified version of the circle method.
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